Topic E.4 — Fission — is the IB Physics nuclear topic with the strongest real-world hook. Half of the marks come from describing reactor components (moderator, control rods, fuel, coolant, containment) and the other half from quantitative calculations of fission energy via mass defect and $E = \Delta m c^2$. The HL extension layers in critical mass and multiplication-factor reasoning, and Paper 2 questions almost always close with an evaluation of nuclear power's environmental trade-offs.
This cheatsheet condenses every formula, reaction equation, trick and trap from Topic E.4 SL + HL onto one page. Scroll to the bottom for the printable PDF, the full notes pack and the gated tutorial library used by Photon Academy students in Singapore.
§1 — Nuclear Fission E.4 SL & HL
Definition
Fission: a heavy nucleus absorbs a slow (thermal) neutron and splits into two lighter nuclei (fission fragments) plus 2–3 neutrons and a large amount of energy.
$$^{1}_{0}n + \,^{235}_{92}\text{U} \rightarrow \,^{141}_{56}\text{Ba} + \,^{92}_{36}\text{Kr} + 3\,^{1}_{0}n$$
Conservation laws
In every nuclear reaction:
- Nucleon number $A$ is conserved.
- Proton number $Z$ is conserved.
- Mass-energy is conserved (the missing mass appears as kinetic energy of the products).
Binding energy & fission
Fission releases energy because the products have higher BE/nucleon than U-235.
U-235: BE/A $\approx 7.6$ MeV. Fragments: BE/A $\approx 8.4$ MeV. $\Rightarrow E \approx 200$ MeV per fission.
§2 — Chain Reactions E.4 SL & HL
Multiplication factor $k$
- $k < 1$: subcritical — reaction dies out.
- $k = 1$: critical — steady-state reactor.
- $k > 1$: supercritical — exponential growth (bomb behaviour).
Critical mass (HL)
Critical mass = minimum mass for a sustained chain reaction ($k \geq 1$). It is determined by a competition between two effects:
- Fission rate $\propto$ volume $\propto r^3$.
- Neutron leakage $\propto$ surface area $\propto r^2$.
Larger mass $\Rightarrow$ smaller surface-to-volume ratio $\Rightarrow$ smaller leakage fraction $\Rightarrow$ chain reaction can be sustained.
§3 — Nuclear Reactor Components E.4 SL & HL
| Component | Function | Example materials |
|---|---|---|
| Fuel rods | Contain the fissile material; site of fission | Enriched U-235 ($\sim$3–5%) |
| Moderator | Slows fast neutrons to thermal energies via elastic collisions | Water (H$_2$O), heavy water (D$_2$O), graphite |
| Control rods | Absorb neutrons to regulate the reaction; fully inserted = shutdown | Boron, cadmium, hafnium |
| Heat exchanger | Transfers thermal energy from coolant to a secondary water circuit (steam) | Pressurised water, CO$_2$ |
| Containment | Shields the environment from radiation | Concrete, lead, steel |
§4 — Energy Calculations E.4 SL & HL (quantitative HL)
Energy density comparison
| Fuel | Energy / MJ kg$^{-1}$ |
|---|---|
| Coal | $\approx 30$ |
| Oil | $\approx 45$ |
| Enriched U (3%) | $\approx 5 \times 10^6$ |
| Pure U-235 | $\approx 8 \times 10^7$ |
Nuclear fuels are roughly $10^5$–$10^6$ times more energy-dense than fossil fuels.
§5 — Safety & Environment E.4 SL & HL
Advantages
- Very high energy density.
- Low CO$_2$ during operation.
- Reliable baseload power (weather-independent).
- Small land footprint.
Disadvantages / risks
- Long-lived radioactive waste (HLW — high-level waste).
- Meltdown / accident risk.
- Thermal pollution (warm water released to aquatic ecosystems).
- High construction cost and long build times.
- Nuclear weapons proliferation risk.
- Uranium mining damage.
Five-step method for fission energy calculations (HL)
- Write the balanced equation and identify all species.
- Find $\Delta m = m_{\text{reactants}} - m_{\text{products}}$ (in u).
- Convert: $E = \Delta m \times 931.5$ MeV (or $\times c^2$ in SI).
- Convert to joules if needed: multiply MeV by $1.602 \times 10^{-13}$.
- Scale to power: $P = R \times E_{\text{fission}}$ or $P = \dot{m}\, c^2$.
Worked Example — Reactor Power Output
Question (HL Paper 2 style — 7 marks)
A pressurised-water reactor (PWR) outputs $P = 1.0$ GW of thermal power from the fission of U-235. Assume each fission releases $E_{\text{fission}} = 200$ MeV.
(a) Calculate the number of fissions per second $R$ in the reactor. [3]
(b) Hence find the rate of mass loss $\dot{m}$ of U-235 (i.e. mass converted to energy per second). [2]
(c) State and explain whether the multiplication factor $k$ is held above, below or exactly at 1 during steady operation. [2]
Solution
- Convert: $E_{\text{fission}} = 200 \times 10^6 \times 1.602 \times 10^{-19} = 3.20 \times 10^{-11}$ J. (M1)
- $R = P / E_{\text{fission}} = (1.0 \times 10^9)/(3.20 \times 10^{-11})$. (M1)
- $R \approx 3.13 \times 10^{19}$ fissions per second. (A1)
- Mass converted: $\dot{m} = P / c^2 = (1.0 \times 10^9)/(3.00 \times 10^8)^2$. (M1)
- $\dot{m} \approx 1.11 \times 10^{-8}$ kg s$^{-1}$ $\approx 11$ μg per second of mass converted to energy. (A1)
- $k = 1$ exactly. (A1)
- Each fission must produce, on average, exactly one further fission — so neutron production matches absorption + leakage. Higher $k$ would be supercritical (power runaway); lower $k$ would shut the reactor down. (R1)
Examiner's note: Two common errors. (i) Confusing the rate of mass conversion ($\dot{m} = P/c^2$, on the order of micrograms per second) with the rate of fuel consumption (kilograms per day) — only a tiny fraction of the fuel mass actually becomes energy. (ii) Saying $k$ "stays close to 1" — for steady operation it must be exactly 1; control rods continuously trim it to that value.
Common Student Questions
Why does fission release energy?
What does the multiplication factor $k$ tell you?
What is the difference between the moderator and the control rods?
Does critical mass mean the mass at which fission starts?
Is nuclear power "pollution-free"?
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