# Nuclear reactions

> IB Physics SL · Unit 5: Nuclear and quantum physics
> Source: https://www.owlsprep.com/study/ib-physics-sl-u5-nuclear-reactions/

This sub-topic covers balancing nuclear reaction equations, mass-energy equivalence for nuclear processes, and the core differences between nuclear fission and fusion. You will learn to calculate energy released from common nuclear reactions.

**Prerequisites:** [Mass-energy equivalence and nuclear structure](https://www.owlsprep.com/study/ib-physics-sl-u5-binding-energy-and-nuclear-structure/); [Radioactive decay fundamentals](https://www.owlsprep.com/study/ib-physics-sl-u5-radioactive-decay/)

## Learning objectives

- Balance nuclear reaction equations for decay, fission, and fusion
- Calculate the Q-value (energy released/absorbed) for nuclear reactions
- Distinguish between nuclear fission and fusion reactions
- Apply mass-energy equivalence to nuclear reaction problems

## Balancing Nuclear Reaction Equations

All nuclear reactions conserve two core quantities across reactants and products: total atomic number (proton number) and total mass number (nucleon number). Conservation of electric charge is automatically satisfied if atomic number is conserved, so you only need to check these two values to balance any reaction.

**Nuclide Notation** — $Z$ = atomic number (number of protons), $A$ = mass number (total number of protons + neutrons), $X$ = chemical symbol of the nuclide.

*Notation:* $_Z^A X$

**Worked example:** Complete the alpha decay reaction of uranium-238: $_{92}^{238}\text{U} \rightarrow _{?}^{?}\text{Th} + _2^4\alpha$

1. Conserve total atomic number across both sides of the reaction

   $$92 = Z + 2 \implies Z = 92 - 2 = 90$$
2. Conserve total mass number across both sides of the reaction

   $$238 = A + 4 \implies A = 238 - 4 = 234$$
3. The resulting thorium nuclide is therefore

   $$_{90}^{234}\text{Th}$$

> **Exam tip:** Always balance both atomic and mass number even if only one is requested — this catches simple arithmetic mistakes.

## Mass Defect and Energy Release (Q-value)

Per Einstein's mass-energy equivalence $E = mc^2$, mass difference between reactants and products in a nuclear reaction is converted to energy. This energy change is called the Q-value of the reaction. A positive Q-value means energy is released (exothermic), while a negative Q-value means energy must be added (endothermic).

**Q-value** — The net energy released or absorbed in a nuclear reaction, calculated from the mass difference between reactants and products.

**Worked example:** Calculate the Q-value for the alpha decay of U-238, given $m_{\text{reactants}} = 238.050788$ u, $m_{\text{products}} = 238.044616$ u. Use 1 u = 931.5 MeV/$c^2$.

1. Calculate the mass difference between reactants and products

   $$\Delta m = m_{\text{reactants}} - m_{\text{products}} = 0.006172 \text{ u}$$
2. Substitute into mass-energy equivalence to find Q

   $$Q = \Delta m c^2 = 0.006172 \text{ u} \times 931.5 \frac{\text{MeV}}{c^2 \cdot \text{u}} \times c^2 \approx 4.2 \text{ MeV}$$
3. The positive Q confirms 4.2 MeV of energy is released in this spontaneous decay.

> **info**
>
> IB exams commonly accept answers in MeV for energy calculations, but always check if the question asks for joules.

## Nuclear Fission

Nuclear fission occurs when a heavy, unstable nucleus absorbs a slow neutron and splits into two smaller lighter nuclei (fission fragments), releasing energy and multiple free neutrons. These extra neutrons can trigger further fission events, creating a self-sustaining chain reaction used in nuclear power reactors.

**Nuclear fission** — A nuclear reaction where a heavy nucleus splits into two lighter nuclei of comparable mass, releasing energy and free neutrons.

*Example:* Fission of uranium-235 in commercial nuclear power plants

**Worked example:** Find the number of neutrons produced in this fission reaction: $_{0}^{1}\text{n} + _{92}^{235}\text{U} \rightarrow _{56}^{141}\text{Ba} + _{36}^{92}\text{Kr} + x_{0}^{1}\text{n}$

1. Check atomic number balance first

   $$\text{Total Z left} = 0 + 92 = 92; \text{Total Z right} = 56 + 36 + 0 = 92$$
2. Balance mass number to solve for x

   $$1 + 235 = 141 + 92 + x(1) \implies 236 = 233 + x \implies x = 3$$
3. This reaction produces 3 free neutrons, which can sustain a chain reaction.

## Nuclear Fusion

Nuclear fusion is the opposite of fission: two light nuclei combine to form a single heavier nucleus. Energy is released because the product nucleus has a higher binding energy per nucleon than the reactant light nuclei. Fusion is the energy source that powers stars like our Sun.

**Nuclear fusion** — A nuclear reaction where two light nuclei fuse to form a single heavier nucleus, releasing large amounts of energy.

*Example:* Proton-proton fusion in the core of the Sun

**Worked example:** Calculate the energy released in deuterium-tritium fusion: $_1^2\text{H} + _1^3\text{H} \rightarrow _2^4\text{He} + _0^1\text{n}$. Given masses: $m_2H = 2.014102$ u, $m_3H = 3.016049$ u, $m_4He = 4.002603$ u, $m_n = 1.008665$ u.

1. Calculate total mass of reactants

   $$m_{\text{reactants}} = 2.014102 + 3.016049 = 5.030151 \text{ u}$$
2. Calculate total mass of products

   $$m_{\text{products}} = 4.002603 + 1.008665 = 5.011268 \text{ u}$$
3. Calculate energy released Q

   $$Q = (5.030151 - 5.011268) \times 931.5 \approx 17.6 \text{ MeV}$$
4. This fusion reaction releases ~4x more energy per kilogram of fuel than fission.

## Common pitfalls

- **Wrong:** Only balance mass number and ignore atomic number
  - Why it fails: Many questions only ask for the mass number of an unknown product, but mistakes in atomic number often lead to wrong mass values anyway
  - Correct: Always balance both atomic number and mass number for every nuclear reaction, even if only one is requested
- **Wrong:** Reverse the mass difference when calculating Q-value
  - Why it fails: Subtracting reactant mass from product mass gives a negative Q for exothermic reactions that release energy
  - Correct: Always use $Q = (m_{\text{reactants}} - m_{\text{products}}) c^2$, so positive Q means energy is released
- **Wrong:** Confuse fission and fusion in exam questions
  - Why it fails: Both release energy, but for different mass ranges, so it is easy to mix up the definitions
  - Correct: Remember: Fission = Fissioning (splitting) a heavy nucleus, Fusion = Fusing (joining) light nuclei
- **Wrong:** Leave energy in MeV when the question asks for joules
  - Why it fails: MeV is convenient for nuclear problems, but exam questions often require SI units for full marks
  - Correct: Convert MeV to joules by multiplying by $1.6 \times 10^{-13}$ J/MeV when requested

## Cheatsheet

| Concept | Key Rule/Value |
| --- | --- |
| Balancing reactions | Conserve atomic number (Z) and mass number (A) |
| Q-value calculation | $Q = (m_{reactants} - m_{products})c^2$, +Q = energy released |
| Unit conversion | 1 u = 931.5 MeV/$c^2$ |
| Nuclear fission | Heavy nucleus splits into two lighter nuclei |
| Nuclear fusion | Two light nuclei join to form one heavier nucleus |

## What's next

Nuclear reactions are the foundation of all applied nuclear physics, from medical imaging to commercial power generation. The balancing and calculation skills you mastered here are required for all subsequent nuclear physics topics, and are regularly tested in both Paper 1 and Paper 2 IB exams. Next, you will explore radioactive decay kinetics and half-life calculations, which rely on your ability to balance decay reactions. This topic also connects to binding energy per nucleon trends, which explain why both fission and fusion release energy, and to energy production topics that cover the environmental impact of nuclear power.

- [Nuclear fission and fusion](https://www.owlsprep.com/study/ib-physics-sl-u5-nuclear-fission-and-fusion/)

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