Apply Fission — Apply E.4 Fission in problem-solving and...

SECTION 1

Historical Context & Motivation

The story of nuclear fission begins in the 1930s, when physicists were still mapping the internal structure of the atom. Scientists had already discovered the proton and neutron, and they knew that enormous amounts of energy were locked inside atomic nuclei. The question was whether that energy could be released in a controlled or useful way. When researchers fired neutrons at uranium, the results were baffling — the products did not match any known pattern of radioactive decay. It took a combination of experimental skill and theoretical insight to realize that the uranium nucleus was actually splitting into two lighter fragments, releasing a staggering amount of energy in the process.

1932

Discovery of the Neutron

James Chadwick identifies the neutron, providing the ideal projectile for probing nuclei because it carries no electric charge and is not repelled by the positive nucleus.

1938

Fission Observed Experimentally

Otto Hahn and Fritz Strassmann bombard uranium with neutrons and detect barium among the products. Lise Meitner and Otto Frisch provide the theoretical explanation, coining the term "fission."

1942

First Self-Sustaining Chain Reaction

Enrico Fermi's team achieves the first controlled, self-sustaining nuclear chain reaction in Chicago Pile-1, demonstrating that fission could be harnessed.

1954

First Nuclear Power Plant

The Obninsk Nuclear Power Plant in the Soviet Union becomes the first to generate electricity for a commercial grid using the energy of nuclear fission.

These breakthroughs raised profound questions that remain central to IB Physics today: How do we calculate the energy released during fission? How does a chain reaction sustain itself? And how can we apply mass–energy equivalence to quantify the output of a nuclear reactor? This lesson equips you with the tools to answer those questions and apply E.4 Fission concepts in problem-solving.

SECTION 2

Core Principles of Nuclear Fission

Nuclear fission is the process in which a heavy nucleus absorbs a neutron and splits into two or more lighter nuclei, releasing additional neutrons and a large quantity of energy. To work confidently with fission problems, you need to understand several foundational ideas that connect nuclear structure to energy release.

Mass Defect & Binding Energy

The mass of a nucleus is always less than the total mass of its individual protons and neutrons. This "missing" mass, called the mass defect (Δm), has been converted into binding energy that holds the nucleus together.

Binding Energy per Nucleon

The binding energy per nucleon curve peaks near iron-56. Nuclei heavier than iron can release energy by splitting into fragments that sit closer to the peak — this is fission.

Chain Reaction

Each fission event releases 2–3 neutrons. If at least one of these neutrons triggers another fission, the process is self-sustaining — a chain reaction.

Critical Mass

A chain reaction only sustains itself when enough fissile material is present so that neutrons are more likely to hit another nucleus than escape. This minimum quantity is called the critical mass.

E = mc²

Einstein's mass–energy equivalence lets us convert the mass defect into energy: E = Δmc². Even a tiny mass difference produces enormous energy because c² ≈ 9 × 10¹⁶ m²·s⁻².

✦ KEY TAKEAWAY

Think of a heavy nucleus like a wobbly water drop balanced on a hilltop. A gentle nudge (a single neutron) can cause it to split into two smaller drops that roll downhill to a more stable state. The "height" it falls corresponds to binding energy per nucleon, and the energy released at the bottom is the kinetic energy of the fission products. The closer the fragments are to the peak of the binding energy curve (near iron), the more energy is released.

SECTION 3

Visualizing the Fission Process

The diagram below illustrates a typical fission event for uranium-235. A slow (thermal) neutron is absorbed by a U-235 nucleus, forming an unstable U-236 compound nucleus. This compound nucleus then splits into two lighter fission fragments, releases 2–3 fast neutrons, and emits gamma radiation. The key point is that the total mass of the products is slightly less than the total mass of the reactants, and that mass difference appears as kinetic energy and radiation.

A thermal neutron strikes a U-235 nucleus, forming an unstable U-236 compound nucleus that splits into barium-141 and krypton-92 (one of many possible fragment pairs), releasing three neutrons and approximately 200 MeV of energy.

Notice in the diagram that the incoming neutron must be slow (thermal) to be absorbed efficiently by U-235. A moderator material (such as water or graphite) slows fast neutrons in a reactor. The released neutrons are fast and must also be slowed before they can trigger subsequent fission events. This is a critical detail for understanding how a controlled chain reaction works, and it frequently appears in IB exam questions.

SECTION 4

Mathematical Framework

Fission problems in IB Physics revolve around a central calculation: finding the energy released from the mass difference between reactants and products. You will use Einstein's mass–energy equivalence and sometimes convert between mass units and energy units. Here are the key equations you need.

MASS–ENERGY EQUIVALENCE

E = Δm × c²

E = energy released (J), Δm = mass defect (kg), c = speed of light = 3.00 × 10⁸ m·s⁻¹. The mass defect is the difference between the total mass of reactants and the total mass of products.

MASS DEFECT

Δm = m(reactants) − m(products)

If Δm > 0, energy is released (exothermic). In fission, the products always have less total mass than the reactants, so energy is liberated.

UNIFIED ATOMIC MASS UNIT CONVERSION

1 u = 931.5 MeV/c²

This shortcut lets you convert mass defect directly from atomic mass units (u) to energy in MeV without converting to kilograms first. If Δm is in u, then E = Δm × 931.5 MeV.

ENERGY PER NUCLEON APPROACH

E(released) = [B.E. per nucleon (products) − B.E. per nucleon (reactant)] × A

An alternative method using the binding energy per nucleon curve. Sum the total binding energies of products and subtract the total binding energy of the parent nucleus. A = mass number. This approach is useful when binding energy per nucleon values are given on the exam.

💡 IB Exam Tip

The IB data booklet provides atomic masses in unified atomic mass units (u) and the conversion 1 u = 931.5 MeV/c². Most fission problems are fastest when you compute Δm in u and multiply by 931.5 to get energy in MeV directly. Only convert to joules if the question specifically asks for SI units (use 1 eV = 1.60 × 10⁻¹⁹ J).

SECTION 5

The Binding Energy Curve and Why Fission Works

The binding energy per nucleon curve is your most powerful visual tool for understanding why fission releases energy. This curve plots the average binding energy per nucleon against mass number A. It rises steeply for light nuclei, peaks near iron-56 (≈ 8.8 MeV per nucleon), and then gradually decreases for heavier elements. Since nuclei "want" to be near the peak (where they are most stable), heavy nuclei like uranium can release energy by splitting into mid-range fragments that have higher binding energy per nucleon.

The binding energy per nucleon curve. Iron-56 sits at the peak. Heavy nuclei (like U-235, far right) release energy by splitting (fission) into fragments closer to the peak. Light nuclei release energy by fusing (fusion). In both cases, the products move upward on this curve, meaning they become more tightly bound.

The energy released per fission event is the difference in total binding energy between the products and the parent nucleus. Since the fission fragments have a higher binding energy per nucleon than uranium, the total binding energy increases when fission occurs. That increase comes at the expense of mass — the products weigh less — and the "lost" mass appears as kinetic energy of the fragments, kinetic energy of the neutrons, and gamma-ray photon energy. Approximately 200 MeV is released per fission of U-235, which is roughly a million times more energy per reaction than a typical chemical reaction.

SECTION 6

Worked Example: Energy Released in U-235 Fission

Let's work through a complete IB-style fission problem step by step. This example uses the mass defect method, which is the most common approach on exams.

📝 Problem Statement

A uranium-235 nucleus absorbs a thermal neutron and undergoes fission according to the reaction: ²³⁵U + ¹n → ¹⁴¹Ba + ⁹²Kr + 3¹n. Given the following atomic masses: m(U-235) = 235.0439 u, m(Ba-141) = 140.9144 u, m(Kr-92) = 91.9262 u, m(n) = 1.0087 u. Calculate the energy released in MeV.

Solution: Energy Released in U-235 Fission

Step 1 — Identify the Reactant MassesThe reactants are one U-235 nucleus and one neutron. Total reactant mass = m(U-235) + m(n) = 235.0439 u + 1.0087 u.

m(reactants) = 236.0526 u

Step 2 — Identify the Product MassesThe products are one Ba-141 nucleus, one Kr-92 nucleus, and three neutrons. Total product mass = m(Ba-141) + m(Kr-92) + 3 × m(n) = 140.9144 u + 91.9262 u + 3 × 1.0087 u = 140.9144 u + 91.9262 u + 3.0261 u.

m(products) = 235.8667 u

Step 3 — Calculate the Mass DefectThe mass defect is the difference between the total reactant mass and total product mass: Δm = m(reactants) − m(products) = 236.0526 u − 235.8667 u.

Δm = 0.1859 u

Step 4 — Convert Mass Defect to EnergyUsing the conversion factor 1 u = 931.5 MeV/c², the energy released is E = Δm × 931.5 MeV = 0.1859 u × 931.5 MeV/u.

E ≈ 173.2 MeV

Step 5 — Interpret the ResultThis 173 MeV represents the kinetic energy of the fission fragments, kinetic energy of the released neutrons, and gamma-ray energy. Note that the total energy released per fission is often quoted as ~200 MeV; the difference comes from additional gamma radiation and beta decay energy of the fission products, which varies by decay channel. Always check whether the question asks for the energy from this specific reaction or the average total energy per fission.

SECTION 7

Controlled vs. Uncontrolled Fission

Not all chain reactions are the same. The distinction between controlled and uncontrolled fission is essential for IB Physics and for understanding the real-world applications and risks of nuclear energy. In a controlled reaction, exactly one neutron from each fission event goes on to cause another fission, maintaining a steady power output. In an uncontrolled reaction, the number of fissions increases exponentially each generation.

Comparison of controlled and uncontrolled nuclear fission

Feature	Controlled Fission (Reactor)	Uncontrolled Fission (Bomb)
Multiplication Factor (k)	k = 1 (critical) — steady state	k > 1 (supercritical) — exponential growth
Moderator	Water or graphite slows neutrons to thermal speeds for efficient fission	No moderator needed; uses highly enriched fuel and fast neutrons
Control Rods	Absorb excess neutrons (e.g., boron or cadmium) to maintain k = 1	No control mechanism — reaction proceeds to completion in microseconds
Fuel Enrichment	Low enrichment: ~3–5% U-235	High enrichment: >80% U-235 or uses Pu-239
Energy Release Rate	Steady and sustained over months/years	Near-instantaneous — enormous energy in a fraction of a second

✦ KEY TAKEAWAY

Think of a controlled chain reaction like a campfire: you add fuel at the right rate and adjust airflow to keep a steady flame. An uncontrolled chain reaction is like throwing all the fuel onto a bonfire at once — the energy release is explosive and impossible to manage. In a reactor, control rods act like the damper on a wood stove, absorbing excess neutrons to keep the reaction at a safe, constant rate.

SECTION 8

Connecting Fission to Fusion and Advanced Topics

IB Physics E.4 often asks you to compare fission with its counterpart, nuclear fusion. Both processes release energy by moving nuclei toward the peak of the binding energy per nucleon curve, but they approach the peak from opposite sides. Understanding this comparison will strengthen your ability to apply the binding energy curve in explanations and earn full marks on extended-response questions.

Fission vs. Fusion comparison

Aspect	Fission	Fusion
Definition	Heavy nucleus splits into lighter fragments	Light nuclei combine to form a heavier nucleus
Typical Fuel	U-235, Pu-239	Deuterium (²H), Tritium (³H)
Energy per Nucleon	~0.8 MeV per nucleon	~3.5 MeV per nucleon (higher)
Conditions Required	Thermal neutron bombardment; achievable at room temperature with a reactor	Extremely high temperature (~10⁷ K) and pressure to overcome Coulomb repulsion
Radioactive Waste	Produces long-lived radioactive fission products	Produces minimal, short-lived radioactive waste
Current Technology	Mature — hundreds of reactors worldwide	Experimental — no sustained net-energy fusion reactor yet

Looking ahead, the IB syllabus connects fission to topics like radioactive decay chains (the fission products are typically unstable and undergo beta decay), nuclear waste management, and the ethical considerations of nuclear technology. At university level, you would encounter concepts like neutron cross-sections (the probability of a nucleus capturing a neutron), reactor criticality calculations, and the role of delayed neutrons in reactor control. For now, mastering the energy calculations and the binding energy curve reasoning will give you a solid foundation for both exams and future study.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Explain, using the binding energy per nucleon curve, why nuclear fission of uranium-235 releases energy while fission of iron-56 does not.

PROBLEM 2 — BASIC CALCULATION

The mass defect for a single fission event of U-235 is 0.186 u. Calculate the energy released in MeV. (Use 1 u = 931.5 MeV/c².)

PROBLEM 3 — INTERMEDIATE

In a nuclear reactor, 2.0 kg of U-235 undergoes complete fission. If each fission event releases 200 MeV, how much total energy is released in joules? (Molar mass of U-235 = 235 g/mol, Avogadro's number = 6.02 × 10²³ mol⁻¹, 1 eV = 1.60 × 10⁻¹⁹ J.)

PROBLEM 4 — APPLIED

A nuclear power plant operates at 1000 MW (thermal) with an efficiency of 33%. The plant uses U-235 fuel, and each fission releases 200 MeV. Calculate the mass of U-235 consumed per day.

PROBLEM 5 — CRITICAL THINKING

In the fission reaction ²³⁵U + ¹n → ¹⁴⁴Ba + ⁸⁹Kr + 3¹n, the binding energy per nucleon values are approximately: U-235 = 7.59 MeV, Ba-144 = 8.26 MeV, Kr-89 = 8.59 MeV. Use the binding energy per nucleon approach (instead of the mass defect approach) to estimate the energy released. Show your working and discuss any discrepancy with the typical ~200 MeV value.

SUMMARY

Lesson Summary

Nuclear fission is the splitting of a heavy nucleus (such as U-235) into two lighter fission fragments upon absorption of a thermal neutron, releasing 2–3 additional neutrons, gamma radiation, and approximately 200 MeV of energy per event. The energy comes from the mass defect — the products have less total mass than the reactants — converted to energy via E = Δmc². Alternatively, the binding energy per nucleon curve explains fission: since fragments lie closer to the peak (near iron-56) than the parent uranium nucleus, the total binding energy increases and the difference is released as kinetic energy.

In problem-solving, calculate the mass defect in u and multiply by 931.5 MeV/u for a quick energy answer. A self-sustaining chain reaction requires critical mass and is controlled in reactors using moderators (to slow neutrons) and control rods (to absorb excess neutrons and maintain k = 1). Compared to fusion, fission releases less energy per nucleon but is technologically mature, making it the basis for all current nuclear power generation.