DP Physics HL E.4 Fission

DP Physics HL E.4 Fission - Full notes

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DP Physics HL | E.4 Fission

DP PHYSICS · THEME E: NUCLEAR AND QUANTUM PHYSICS

E.4

Fission

Full notes for HL · IB-style

Contents

1	Nuclear fission
2	Chain reactions
3	Nuclear reactors
4	Fission as an energy source
5	Data-based questions
6	Exam technique and further practice
7	Common misconceptions and links
8	Exam-style question with markscheme
9	Check your understanding
10	Summary and topic map

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and do every Quick check before moving to the next section.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
describe neutron-induced fission and write balanced fission equations
explain energy release using binding energy per nucleon and mass defect
describe chain reactions, multiplication factor and critical mass
describe the roles of moderator, control rods, coolant and shielding
calculate energy and fuel use for reactors
discuss benefits, risks and waste of nuclear power

There is no additional HL content in this subtopic. SL and HL students study the same material.

Guiding questions

How is energy released in nuclear fission? How is a fission chain reaction controlled in a reactor? What are the advantages and disadvantages of nuclear power?

How this topic is examined

E.4 appears in Papers 1 and 2. Expect fission equations, energy calculations from masses or binding energies, reactor component explanations and evaluation of nuclear power.

Distinguish clearly between moderator and control rods.

Command term	What the examiner wants
State	a brief answer with no explanation
Outline	a brief account or summary
Describe	a detailed account without reasons
Explain	a detailed account including causes, reasons or mechanisms
Calculate	a numerical answer with working and correct units
Determine	obtain the only possible answer, often from a graph or data
Show that	obtain the given result, showing every step (give one more significant figure)
Estimate	an approximate value using reasonable assumptions
Sketch	a graph or diagram showing the general shape and key features
Deduce / Suggest	reach a conclusion from given information; propose a reasonable explanation

How to use these notes

Each section teaches one part of the syllabus in full. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Key point	the one idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
In the real world	where the physics shows up outside the classroom
Try this	something quick you can do yourself to test the idea
Worked example	full solutions with IB-style marks (M1 method, A1 answer, R1 reasoning)
Quick check	short questions to test yourself at the end of each section

1 Nuclear fission

Splitting heavy nuclei.

Think about it

How can a fuel pellet the size of a fingertip release as much energy as a tonne of coal?

Nuclear fission powers reactors that supply about 10% of the world's electricity. A heavy nucleus splits into two lighter ones, releasing energy because the products are more tightly bound.

1.1 What is fission?

In nuclear fission, a heavy nucleus splits into two smaller nuclei of roughly similar mass, together with two or three neutrons and a large release of energy. Fission can occur spontaneously, but in reactors it is induced by absorbing a slow (thermal) neutron.

Figure 1. Neutron-induced fission of uranium-235.

WORKED EXAMPLE 1 Balancing (standard)

Complete: ¹₀n + ²³⁵₉₂U → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + x ¹₀n. Find x.

Solution

Nucleons: 1 + 235 = 141 + 92 + xM1

x = 3A1

Quick check

1. Check that charge is conserved in the reaction above.

Answers: 1. 92 = 56 + 36

1.2 Discovering fission

In 1938 Hahn and Strassmann found barium among the products of bombarding uranium with neutrons. Meitner and Frisch explained that the uranium nucleus had split, and used E = mc² to estimate the 200 MeV released. Within four years Fermi's team built the first controlled chain reaction (1942).

Quick check

1. What unexpected product showed that uranium nuclei had split?

Answers: 1. barium, a much lighter element

1.3 Fission products

The fragments are not always the same: U-235 can split in dozens of different ways. Fission fragments have too many neutrons for their mass and are highly radioactive, decaying by beta-minus emission. The yield peaks for fragments of mass about 95 and 139.

Figure 2. Fission-fragment yield against mass number for U-235.

WORKED EXAMPLE 2 Another split (standard)

U-235 absorbs a neutron and splits into xenon-140 (Z = 54) and strontium-94, releasing two neutrons. Find Z for strontium.

Solution

92 = 54 + ZM1

Z = 38A1

1.4 Why fission releases energy

Binding energy per nucleon is about 7.6 MeV for uranium but about 8.5 MeV for the medium-mass fragments. The products are more tightly bound, so the total binding energy increases and the difference is released, mostly as the kinetic energy of the fragments.

Figure 3. U-236 splitting into Ba-141 and Kr-92 moves up the binding energy curve.

WORKED EXAMPLE 3 From binding energies (exam level)

Estimate the energy released when U-236 (7.59 MeV per nucleon) splits into Ba-141 (8.38 MeV per nucleon) and Kr-92 (8.66 MeV per nucleon) plus 3 neutrons.

Solution

BE after = 141 × 8.38 + 92 × 8.66 = 1978 MeV; before = 236 × 7.59 = 1791 MeVM1

Energy released ≈ 187 MeVA1

1.5 Calculating energy from masses

The energy released equals the mass decrease times c², or Δm × 931.5 MeV u⁻¹.

Figure 4. Total mass before and after the fission reaction.

Key point

Typical energy per U-235 fission: about 200 MeV, mostly as kinetic energy of the two fragments.

WORKED EXAMPLE 4 From masses (exam level)

Masses: U-235 235.0439 u, n 1.0087 u, Ba-141 140.9144 u, Kr-92 91.9262 u. Find the energy released in the reaction with 3 neutrons emitted.

Solution

Δm = (235.0439 + 1.0087) − (140.9144 + 91.9262 + 3 × 1.0087) = 0.1859 uM1 A1

E = 0.1859 × 931.5 = 173 MeVA1

WORKED EXAMPLE 5 In joules (standard)

Convert 200 MeV into joules.

Solution

200 × 10⁶ × 1.6 × 10⁻¹⁹M1

= 3.2 × 10⁻¹¹ JA1

1.6 Where the energy goes

Most of the energy appears as kinetic energy of the fission fragments (about 170 MeV), which collide with surrounding atoms and heat the fuel. The rest is carried by neutrons, prompt gamma rays and later beta decays of the products (including energy carried away by antineutrinos).

Form	Approximate energy / MeV
kinetic energy of fragments	170
kinetic energy of neutrons	5
prompt gamma rays	7
beta and gamma from products	15
antineutrinos (lost)	10

Quick check

1. Which form of energy is not captured in a reactor?

Answers: 1. antineutrino energy

2 Chain reactions

One fission triggering many.

2.1 Chain reactions

Each fission releases two or three neutrons, which can induce further fissions. If on average more than one neutron per fission causes another fission, the number of fissions grows rapidly (an uncontrolled chain reaction). If exactly one does, the rate is steady (controlled).

Figure 5. A branching chain reaction.

Quick check

1. What happens if on average fewer than one neutron per fission causes another fission?

Answers: 1. the reaction dies out

2.2 Multiplication factor

The multiplication factor k is the average number of neutrons from each fission that cause another fission. k > 1: supercritical (growing); k = 1: critical (steady); k < 1: subcritical (dying out). A reactor is held at k = 1.

Figure 6. Number of fissions per generation for different multiplication factors.

WORKED EXAMPLE 6 Growth (standard)

Starting from 1 fission with k = 2, how many fissions occur in the 20th generation?

Solution

2²⁰M1

≈ 10⁶A1

2.3 Critical mass

The critical mass is the minimum mass of fissile material for a self-sustaining chain reaction. A small lump loses too many neutrons through its surface; a larger lump has a smaller surface-to-volume ratio, so more neutrons cause fissions before escaping. Shape, density, purity and neutron reflectors all affect it.

WORKED EXAMPLE 7 Surface to volume (exam level)

Explain why doubling the radius of a sphere of fuel makes a chain reaction more likely.

Solution

Volume (fissions) increases by 8; surface (escape) by 4M1

Surface-to-volume ratio halves, so a smaller fraction of neutrons escapesR1

2.4 The neutron life cycle

Of the roughly 2.4 neutrons from each fission, some escape the core, some are absorbed by U-238, the moderator, coolant or control rods, and in a steady reactor exactly one goes on to cause another fission.

Figure 7. What happens to the neutrons from one fission in a steady reactor.

Quick check

1. How does a neutron reflector around the core help?

Answers: 1. it returns escaping neutrons, reducing the critical mass

2.5 Delayed neutrons and control

About 0.65% of fission neutrons are released seconds after fission by decaying fragments (delayed neutrons). Because the chain reaction depends slightly on them, the power changes slowly enough for control rods to respond. Without them, reactors would be almost impossible to control.

Figure 8. Reactor power responding to raising and lowering the control rods (schematic).

Quick check

1. Why are delayed neutrons essential for reactor control?

Answers: 1. they slow the rate of change of power so mechanical rods can keep up

2.6 Surface-to-volume ratio

For a sphere, surface area/volume = 3/r, so it falls as the radius grows. Neutrons are produced throughout the volume but escape through the surface, so larger lumps lose a smaller fraction of their neutrons.

Figure 9. Surface-to-volume ratio of a sphere against radius.

Quick check

1. How does the surface-to-volume ratio change when the radius doubles?

Answers: 1. it halves

2.7 Fissile and fertile materials

Uranium-235 undergoes fission with slow neutrons and is fissile, but natural uranium is only 0.72% U-235. Uranium-238 rarely undergoes fission but can absorb a neutron and, after two beta decays, become plutonium-239, which is fissile. Fuel for most reactors is enriched to 3–5% U-235.

Figure 10. U-235 content of natural uranium, reactor fuel and weapons-grade uranium.

WORKED EXAMPLE 8 Breeding (exam level)

Write the equations for U-238 capturing a neutron and decaying by two β⁻ emissions.

Solution

²³⁸₉₂U + ¹₀n → ²³⁹₉₂UA1

²³⁹₉₂U → ²³⁹₉₃Np + e⁻ + ν̅ → ²³⁹₉₄Pu + e⁻ + ν̅A1

3 Nuclear reactors

Controlling the chain reaction.

Think about it

Water in a reactor slows neutrons down. Why do we want slower neutrons?

3.1 Main parts of a thermal reactor

A thermal fission reactor contains fuel rods, a moderator, control rods, a coolant, a heat exchanger and heavy shielding.

Component	Function
fuel rods	contain enriched uranium (or plutonium)
moderator	slows fast neutrons to thermal speeds
control rods	absorb neutrons to control the reaction rate
coolant	carries thermal energy to the heat exchanger
heat exchanger	transfers energy to make steam for turbines
shielding	absorbs radiation to protect workers and environment

Figure 11. A pressurised water reactor (schematic).

3.2 The moderator

Fast neutrons (about 2 MeV) from fission are unlikely to cause further U-235 fissions; slow (thermal) neutrons, about 0.025 eV, are far more likely to be absorbed. The moderator slows neutrons by elastic collisions. Light nuclei such as hydrogen in water or carbon in graphite work best because a neutron transfers a large fraction of its kinetic energy in each collision.

Figure 12. A neutron transfers much of its energy to a light nucleus of similar mass.

Figure 13. Neutron energy against number of collisions in water and graphite (average values).

WORKED EXAMPLE 9 Head-on collision (exam level)

Show that a neutron colliding elastically head-on with a stationary proton (equal mass) transfers all its kinetic energy.

Solution

Momentum and kinetic energy conserved with equal massesM1

Velocities are exchanged: neutron stops, proton moves offR1

WORKED EXAMPLE 10 Thermal speed (standard)

Find the speed of a neutron of energy 0.025 eV (m = 1.67 × 10⁻²⁷ kg).

Solution

v = √(2E/m) = √(2 × 4.0 × 10⁻²¹/1.67 × 10⁻²⁷)M1

= 2200 m s⁻¹A1

Common mistake

Thinking the moderator slows the reaction down. It slows the neutrons, making fission more likely; the control rods control the rate.

3.3 Why slow neutrons?

The probability that a neutron causes fission of U-235 is several hundred times larger for thermal neutrons than for fast fission neutrons. Thermal reactors therefore use a moderator; fast reactors use more highly enriched fuel and no moderator.

Figure 14. Approximate fission probability of U-235 against neutron energy (log scales).

Quick check

1. Why is no moderator needed in a fast breeder reactor?

Answers: 1. it uses highly enriched fuel (or plutonium) that fissions with fast neutrons

3.4 Control rods

Control rods made of boron or cadmium absorb neutrons. Lowering them reduces the number of neutrons available for fission (k falls); raising them increases it. In an emergency, rods drop fully in (a 'scram') to shut down the chain reaction.

Quick check

1. What happens to the reactor power if the control rods are raised slightly?

Answers: 1. k rises above 1 briefly and the power increases

3.5 Types of reactor

Reactor	Moderator	Coolant	Notes
pressurised water (PWR)	water	water at high pressure	most common worldwide
boiling water (BWR)	water	water boiling in core	steam goes directly to turbine
CANDU	heavy water	heavy water	can use natural uranium
advanced gas-cooled	graphite	carbon dioxide	used in the UK
fast breeder	none	liquid sodium	breeds plutonium from U-238

Quick check

1. Why can CANDU reactors use natural uranium?

Answers: 1. heavy water absorbs very few neutrons

3.6 Decay heat after shutdown

When a reactor shuts down, fission stops but the radioactive fission products continue to decay, releasing heat: about 6–7% of full power at once, about 1% after a few hours and still a fraction of a percent after a week. Cooling must continue for months.

Figure 15. Decay heat against time after shutdown (approximate).

WORKED EXAMPLE 11 Decay heat (exam level)

A 3000 MW (thermal) reactor shuts down. Estimate the decay heat after one hour, when it is about 1.3% of full power.

Solution

0.013 × 3000 MWM1

≈ 39 MW: enough to melt the core without coolingA1

3.7 Coolant and heat exchanger

The coolant (pressurised water, gas or liquid sodium) carries thermal energy from the core. In a heat exchanger it boils water in a separate circuit; the steam drives turbines connected to generators, then is condensed and returned.

WORKED EXAMPLE 12 Coolant (exam level)

A reactor transfers 3000 MW to pressurised water whose temperature rises from 290 °C to 325 °C (c = 5.5 kJ kg⁻¹ K⁻¹ at this temperature). Find the mass flow rate.

Solution

P = (m/t)cΔTM1

m/t = 3.0 × 10⁹/(5500 × 35) = 1.6 × 10⁴ kg s⁻¹A1

3.8 Energy flow in a power station

About a third of the thermal energy from fission becomes electrical energy; the rest heats cooling water and the surroundings, as required by the second law of thermodynamics.

Figure 16. Sankey diagram for a nuclear power station.

WORKED EXAMPLE 13 Fuel use (exam level)

A 1000 MW (electrical) power station is 33% efficient. Each fission releases 200 MeV. Find the number of fissions per second and the mass of U-235 used per day.

Solution

Thermal power = 3.0 × 10⁹ W; fissions = 3.0 × 10⁹/3.2 × 10⁻¹¹ = 9.4 × 10¹⁹ s⁻¹M1 A1

Mass per day = 9.4 × 10¹⁹ × 86 400 × 235 × 1.66 × 10⁻²⁷ = 3.2 kgA1

3.9 Why only one third?

A PWR's steam is limited to about 300 °C (to keep the pressurised water liquid), so the maximum possible (Carnot) efficiency is only about 45%, and real stations achieve about 33%. Gas turbines run hotter and reach higher efficiencies.

Figure 17. Typical efficiencies of power stations.

WORKED EXAMPLE 14 Carnot limit (exam level)

Estimate the Carnot efficiency for steam at 300 °C and cooling water at 30 °C.

Solution

η = 1 − 303/573M1

= 47%A1

4 Fission as an energy source

Benefits, risks and waste.

4.1 Energy density

Nuclear fuel has an enormous energy density: complete fission of 1 kg of U-235 releases about 8 × 10¹³ J, roughly 3 million times more than burning 1 kg of coal.

Figure 18. Energy densities of fuels (logarithmic scale).

WORKED EXAMPLE 15 Energy density (exam level)

Calculate the energy released by fission of 1.0 kg of U-235 (200 MeV per fission).

Solution

N = 1000/235 × 6.02 × 10²³ = 2.56 × 10²⁴M1

E = 2.56 × 10²⁴ × 3.2 × 10⁻¹¹ = 8.2 × 10¹³ JA1

4.2 Fission and fusion compared

Both fission and fusion release energy by moving nuclei towards the peak of the binding energy curve: fission by splitting heavy nuclei, fusion by joining light nuclei (E.5). Fusion releases more energy per nucleon, but needs extreme temperatures to overcome the electric repulsion of the nuclei.

Figure 19. Fission and fusion both move nuclei towards iron-56 on the binding energy curve.

Quick check

1. Why can iron-56 not release energy by either fission or fusion?

Answers: 1. it is at the peak of the curve: any change lowers binding energy per nucleon

4.3 Reactor output over a year

Nuclear stations run at nearly constant output for long periods, stopping only for refuelling and maintenance. Their capacity factor (actual output/maximum possible) is typically above 90%, much higher than for wind or solar.

Figure 20. Monthly average output of a 1.2 GW nuclear station with a refuelling outage.

WORKED EXAMPLE 16 Capacity factor (exam level)

Use the graph to estimate the station's capacity factor for the year.

Solution

Average output = sum/12 = 1.05 GWM1

Capacity factor = 1.05/1.2 = 87%A1

4.4 Advantages and disadvantages

Advantages	Disadvantages
very high energy density	radioactive waste with long half-lives
no CO₂ during operation	risk of accidents (Chernobyl, Fukushima)
reliable baseload power	high construction and decommissioning costs
small land area	uranium is finite; mining has impacts
fuel reserves for many decades	links to nuclear weapons proliferation

Quick check

1. Give one argument for and one against building new nuclear power stations.

Answers: 1. for: low-carbon reliable power; against: long-lived waste or accident risk

4.5 Nuclear waste

Spent fuel contains fission products (strongly radioactive, half-lives mostly up to tens of years) and transuranic elements such as plutonium (half-lives of thousands of years). It is stored in cooling ponds, then sealed in glass and steel for long-term geological disposal.

Level	Examples	Treatment
high-level	spent fuel, fission products	cooled, vitrified, deep geological storage
intermediate	reactor components, fuel cladding	encased in concrete, stored underground
low-level	protective clothing, tools	compacted, shallow burial

Figure 21. Approximate activity of spent fuel against time (log scales).

4.6 Safety features

Reactors use several barriers (fuel cladding, pressure vessel, containment building), negative feedback (many reactors become less reactive if they overheat), automatic shutdown systems, and emergency cooling to remove the decay heat that continues after shutdown.

In the real world

About 7% of a reactor's power continues as decay heat immediately after shutdown. The Fukushima accident (2011) happened when cooling failed after the tsunami, even though the chain reactions had stopped.

Quick check

1. Why must reactors be cooled even after shutdown?

Answers: 1. radioactive fission products continue to release decay heat

4.7 Enrichment

Natural uranium is converted into uranium hexafluoride gas and spun in thousands of gas centrifuges. The slightly lighter U-235 molecules concentrate near the axis; repeating the process in cascades raises the U-235 content from 0.72% to 3–5%.

Figure 22. A gas centrifuge separates uranium isotopes by mass.

Quick check

1. Why is enrichment slow and costly?

Answers: 1. the masses of the isotopes differ by less than 1%, so each stage separates them only slightly

4.8 Accidents and lessons

Chernobyl (1986) combined an unstable reactor design with unsafe testing; a power surge destroyed the reactor and spread radioactive material across Europe. At Fukushima (2011), a tsunami knocked out cooling, and decay heat led to meltdowns and hydrogen explosions. Both led to stricter safety standards worldwide.

Quick check

1. What caused the meltdowns at Fukushima, given that the chain reactions had stopped?

Answers: 1. loss of cooling for decay heat

4.9 Nuclear power around the world

The share of electricity from nuclear power varies enormously between countries, reflecting decisions about energy security, cost, resources and public opinion.

Figure 23. Approximate share of electricity generated by nuclear power in selected countries.

Quick check

1. Suggest one reason a country might choose a large nuclear share.

Answers: 1. few fossil fuel resources; energy security; low-carbon targets

4.10 Small modular reactors and the future

Small modular reactors (SMRs) are factory-built units of up to about 300 MW, designed to be cheaper and quicker to build, with passive safety systems that cool the core by natural circulation. Research also continues on fast reactors that burn long-lived waste and on thorium fuel cycles.

WORKED EXAMPLE 17 Passive safety (standard)

Explain what is meant by passive safety and why it matters after Fukushima.

Solution

Cooling or shutdown happens through natural processes (gravity, convection) without pumps or powerR1

Decay heat is removed even if electrical supplies failR1

WORKED EXAMPLE 18 Thorium (exam level)

Thorium-232 absorbs a neutron and undergoes two β⁻ decays. Identify the product and explain why it is useful.

Solution

²³³₉₂U (uranium-233)A1

It is fissile, so thorium can be used to breed reactor fuelR1

4.11 Comparing energy sources

Source	Lifecycle CO₂ / g per kWh	Reliability
coal	about 820	high
natural gas	about 490	high
solar PV	about 40	variable
wind	about 11	variable
nuclear	about 12	high (baseload)

Quick check

1. Why is nuclear power described as low-carbon but not zero-carbon?

Answers: 1. mining, enrichment and construction produce emissions

5 Data-based questions

Interpreting fission data.

5.1 Data: reactor output and fuel burn-up

A power station records its electrical output over a year and the fuel it uses.

Quantity	Value
average electrical output	1.2 GW
efficiency	34%
energy per fission	200 MeV
fuel enrichment	4.0% U-235
fraction of U-235 used before refuelling	about 75%

WORKED EXAMPLE 19 Thermal power (standard)

Find the average thermal power.

Solution

1.2/0.34M1

= 3.5 GWA1

WORKED EXAMPLE 20 U-235 per year (exam level)

Find the mass of U-235 undergoing fission per year.

Solution

Fissions per second = 3.5 × 10⁹/3.2 × 10⁻¹¹ = 1.1 × 10²⁰M1

Mass = 1.1 × 10²⁰ × 3.16 × 10⁷ × 3.9 × 10⁻²⁵ kg = 1360 kgA1

WORKED EXAMPLE 21 Fuel per year (exam level)

Estimate the mass of enriched fuel loaded per year.

Solution

U-235 loaded = 1360/0.75 = 1810 kgM1

Fuel = 1810/0.040 = 45 tonnesA1

5.2 Data: comparing fuels

Fuel	Energy density / MJ kg⁻¹	Mass for 1 GW for a year
coal	30	about 3 million tonnes (at 35%)
natural gas	55	about 1 million tonnes (at 55%)
enriched uranium (4%)	about 3 × 10⁶ (burn-up)	about 30 tonnes

WORKED EXAMPLE 22 Coal check (exam level)

Verify the coal figure: 1 GW electrical at 35% efficiency for one year using coal at 30 MJ kg⁻¹.

Solution

Thermal energy = (1.0 × 10⁹/0.35) × 3.16 × 10⁷ = 9.0 × 10¹⁶ JM1

Mass = 9.0 × 10¹⁶/3.0 × 10⁷ = 3.0 × 10⁹ kg (3 million tonnes)A1

6 Exam technique and further practice

Applying the ideas.

6.1 Method checklist

•Balance nucleon and proton numbers in fission equations.
•Energy from masses: Δm × 931.5 MeV u⁻¹.
•Explain the moderator as slowing neutrons, control rods as absorbing them.
•Use 200 MeV per fission and Avogadro's number for fuel calculations.
•Link energy release to increased binding energy per nucleon.

6.2 Multi-step problem: reactor fuel

WORKED EXAMPLE 23 Annual fuel (exam level)

A reactor produces 3.0 GW of thermal power. Find (a) the fissions per second; (b) the mass of U-235 used per year; (c) the mass of enriched fuel (4% U-235) needed per year.

Solution

(a) 3.0 × 10⁹/3.2 × 10⁻¹¹ = 9.4 × 10¹⁹ s⁻¹A1

(b) 9.4 × 10¹⁹ × 3.16 × 10⁷ × 3.9 × 10⁻²⁵ kg = 1150 kgM1 A1

6.3 Writing an evaluation of nuclear power

Extended questions often ask you to evaluate nuclear power. A strong answer gives balanced points with physics behind them, uses numbers where possible, and reaches a justified conclusion.

•Energy density: about 10⁶ times that of fossil fuels, so tiny fuel volumes.
•Emissions: very low lifecycle CO₂ (about 12 g per kWh).
•Reliability: steady baseload output, unlike wind and solar.
•Waste: long-lived isotopes need secure storage for thousands of years.
•Safety: low accident rate, but severe consequences when accidents occur; decay heat must always be removed.
•Cost and time: high construction costs and long build times.

WORKED EXAMPLE 24 Evaluation (exam level)

Evaluate the use of nuclear fission to reduce carbon emissions from electricity generation.

Solution

Low lifecycle CO₂ and reliable baseload output, with very high energy densityR1

Against: long-lived waste, accident risk and high cost; uranium is finiteR1

Justified conclusion, for example, useful alongside renewables if waste is managedR1

6.4 Multi-step problem: moderation and collisions

WORKED EXAMPLE 25 Graphite moderator (exam level)

In a head-on elastic collision with a stationary carbon-12 nucleus, a neutron keeps (11/13)² of its kinetic energy. Find (a) this fraction; (b) its energy after one collision if it starts at 2.0 MeV; (c) the minimum number of head-on collisions to reach below 1 eV.

Solution

(a) (11/13)² = 0.716A1

(b) 1.43 MeVA1

6.5 Graded practice

WORKED EXAMPLE 26 Missing nucleus (standard)

Complete: ¹₀n + ²³⁵₉₂U → ¹³⁷₅₂Te + X + 2 ¹₀n.

Solution

A = 236 − 137 − 2 = 97; Z = 92 − 52 = 40M1

X = ⁹⁷₄₀ZrA1

WORKED EXAMPLE 27 Mass to energy (standard)

A fission reaction has a mass decrease of 0.20 u. Find the energy released in MeV and J.

Solution

0.20 × 931.5 = 186 MeVA1

= 3.0 × 10⁻¹¹ JA1

WORKED EXAMPLE 28 Neutron speed (standard)

A fast neutron has 2.0 MeV of kinetic energy. Find its speed.

Solution

v = √(2 × 3.2 × 10⁻¹³/1.67 × 10⁻²⁷)M1

= 2.0 × 10⁷ m s⁻¹A1

6.6 Graded practice: energy calculations

WORKED EXAMPLE 29 Per nucleon (standard)

A fission releases 200 MeV from 236 nucleons. Find the energy per nucleon and compare with combustion (about 4 eV per atom).

Solution

200/236 = 0.85 MeV per nucleonA1

About 10⁵ times the energy per atom of combustionA1

WORKED EXAMPLE 30 Power from fission rate (standard)

A reactor core undergoes 6.0 × 10¹⁹ fissions per second at 200 MeV each. Find the thermal power.

Solution

P = 6.0 × 10¹⁹ × 3.2 × 10⁻¹¹M1

= 1.9 GWA1

WORKED EXAMPLE 31 Equivalent coal (exam level)

Coal releases 30 MJ kg⁻¹. What mass of coal releases the same energy as 1.0 g of U-235 undergoing fission?

Solution

1.0 g U-235: 8.2 × 10¹⁰ JM1

Coal mass = 8.2 × 10¹⁰/3.0 × 10⁷ = 2.7 tonnesA1

6.7 Graded practice: reactors

WORKED EXAMPLE 32 Collision energy loss (exam level)

On average a neutron loses about 63% of its energy per collision with hydrogen. Estimate the collisions to slow it from 2.0 MeV to 0.025 eV.

Solution

(0.37)ⁿ = 0.025/2.0 × 10⁶ = 1.25 × 10⁻⁸M1

n = ln(1.25 × 10⁻⁸)/ln 0.37 ≈ 18 collisionsA1

WORKED EXAMPLE 33 Rods inserted (standard)

State and explain what happens to k and to the reactor power when control rods are pushed further in.

Solution

More neutrons absorbed: k < 1R1

Fission rate and power fallR1

WORKED EXAMPLE 34 Shielding (standard)

Why is the core surrounded by thick concrete?

Solution

Absorbs neutrons and gamma radiationR1

Protects workers and the environmentR1

6.8 Multi-step problem: binding energy and products

WORKED EXAMPLE 35 Strontium–xenon split (exam level)

U-236 (7.59 MeV per nucleon) splits into Sr-94 (8.59 MeV per nucleon), Xe-140 (8.29 MeV per nucleon) and 2 neutrons. Find (a) the total binding energy before; (b) after; (c) the energy released and the mass decrease.

Solution

(a) 236 × 7.59 = 1791 MeVA1

(b) 94 × 8.59 + 140 × 8.29 = 1968 MeVA1

6.9 Graded practice: chain reactions

WORKED EXAMPLE 36 Generation time (exam level)

In a bomb, each generation takes about 10 ns and k = 2. Estimate how long it takes to go from one fission to 10²⁴ fissions in a single generation.

Solution

2ⁿ = 10²⁴: n = 80M1

t ≈ 80 × 10 ns = 0.8 μsA1

WORKED EXAMPLE 37 Subcritical (standard)

A sample has k = 0.9. Starting with 1000 fissions, how many occur in the 10th generation?

Solution

1000 × 0.9¹⁰M1

≈ 350: the reaction dies awayA1

6.10 Graded practice: explanations

WORKED EXAMPLE 38 Why neutrons (exam level)

Explain why neutrons, not protons, are used to induce fission.

Solution

Neutrons are unchargedR1

So they are not repelled by the nucleus and can enter it even at low speedR1

WORKED EXAMPLE 39 Fragments radioactive (standard)

Explain why fission fragments are usually beta-minus emitters.

Solution

Heavy nuclei have a high N/Z ratio; fragments inherit itR1

Fragments are neutron-rich for their size: β⁻ decay converts neutrons to protonsR1

WORKED EXAMPLE 40 Water as moderator and coolant (standard)

Give two reasons water is used in PWRs.

Solution

Hydrogen nuclei slow neutrons efficientlyR1

High specific heat capacity: good coolant; cheap and plentifulR1

6.11 Graded practice: mixed

WORKED EXAMPLE 41 Plutonium fission (standard)

Complete: ¹₀n + ²³⁹₉₄Pu → ¹⁴⁵₅₆Ba + ⁹²₃₈Sr + x ¹₀n.

Solution

240 = 145 + 92 + xM1

x = 3A1

WORKED EXAMPLE 42 Thermal neutron (standard)

Find the kinetic energy in joules of a neutron moving at 2200 m s⁻¹.

Solution

E = ½(1.67 × 10⁻²⁷)(2200)²M1

= 4.0 × 10⁻²¹ J (0.025 eV)A1

6.12 Self-test 2

Quantity	Equation or value
mass–energy	E = Δmc²; 1 u = 931.5 MeV
energy per U-235 fission	about 200 MeV = 3.2 × 10⁻¹¹ J
number of nuclei in mass m	N = (m/M) × NA
power from fission rate	P = (fissions per second) × (energy per fission)
thermal neutron energy	about 0.025 eV (2200 m s⁻¹)
fast neutron energy	about 2 MeV
efficiency	electrical output / thermal input

Term	Meaning
fission	splitting of a heavy nucleus into two smaller nuclei
chain reaction	neutrons from one fission cause further fissions
critical mass	minimum mass for a self-sustaining chain reaction
moderator	material that slows neutrons
control rods	neutron absorbers that control the reaction rate
enrichment	increasing the proportion of U-235
thermal neutron	slow neutron of energy about 0.025 eV

Students often think…	Actually…
the moderator controls the reaction rate	it slows neutrons; control rods control the rate
fission releases energy because mass is destroyed	binding energy increases; mass decrease corresponds to energy released
fast neutrons are best for causing fission of U-235	thermal neutrons are far more effective
a nuclear reactor can explode like a bomb	fuel enrichment and design prevent a nuclear explosion
nuclear power emits no CO₂	construction and mining do

Topic	Connection
E.3 Radioactive decay	binding energy, half-life of waste
A.3 Work, energy and power	efficiency and power
B.4 Thermodynamics (HL)	heat engine efficiency
A.2 Forces and momentum	elastic collisions in the moderator

Idea	Remember
Fission	heavy nucleus splits; 2–3 neutrons
Energy	≈ 200 MeV per fission; binding energy increase
Chain reaction	k > 1, k = 1, k < 1
Critical mass	minimum for sustained reaction
Reactor	fuel, moderator, control rods, coolant
Issues	waste, safety, CO₂

Part	Answer	Marks
(a)	1 + 235 = 141 + 92 + 3 = 236	A1
	92 = 56 + 36	A1
(b)	products have greater binding energy per nucleon	R1
	difference released as kinetic energy of fragments	R1
(c)	173 × 1.6 × 10⁻¹³ = 2.8 × 10⁻¹¹ J	A1
(d)	slows fast neutrons by collisions	R1
	slow neutrons more likely to cause fission	R1
(e)	absorb neutrons	R1
	adjusted so on average one neutron per fission causes another (k = 1)	R1