Nuclear Equations and Conservation

Branch note: This page deepens one part of Nuclear Physics.

9749 scope

Core work uses simple nuclear equations, conservation of nucleon number, charge and total mass-energy, plus the qualitative prediction of a neutrino from beta-decay energy and momentum evidence. Positron emission is not required; antineutrino and antiparticle knowledge is not required. Fuller particle equations are retained only as enrichment.

Overview

Nuclear equations describe changes in nuclei using nuclide notation. They are constrained by conservation laws, even when the detailed nuclear mechanism is complicated.

This branch collects the nuclear-specific conservation ideas needed for:

• Radioactive Decay
• Beta-Decay Particle Context
• Nuclear Fission
• Nuclear Fusion

The broader cross-topic treatment remains in Conservation Laws in Physics.

Core Ideas

• The top number $A$ is nucleon number and must balance in nuclear equations.
• The bottom number $Z$ is proton number for nuclei, but across the whole equation it acts as charge number.
• Alpha decay decreases $A$ by 4 and $Z$ by 2.
• Beta-minus decay keeps $A$ unchanged and increases nuclear $Z$ by 1.
• Enrichment: beta-plus decay keeps $A$ unchanged and decreases nuclear $Z$ by 1; it is not required by 9749.
• Gamma emission changes nuclear energy state but not $A$ or $Z$.
• Rest mass need not be separately conserved; mass-energy is conserved.
• Momentum is conserved, so nuclear products recoil.

Exam Relevance

Most exam questions test whether students can balance nuclear equations, identify missing particles or daughter nuclei, and explain energy release without saying that energy is created.

Definition

A nuclear equation is a symbolic representation of a nuclear transformation in which the total nucleon number and total charge number must balance.

Why It Matters

Conservation laws prevent impossible nuclear equations. They also explain why an additional neutral particle was predicted in beta decay and why fission or fusion can release energy without energy being created.

Key Representations

Nuclide Notation

A nucleus is written as:

$${}_Z^{A}X$$

where:

• $A$ = nucleon number
• $Z$ = proton number
• $X$ = chemical symbol

Neutron number is:

$$N=A-Z$$

Conservation Checklist

For a nuclear equation, check:

1. total top numbers balance
2. total bottom numbers balance
3. charge is conserved
4. energy is conserved when rest energy, kinetic energy, and radiation are counted
5. momentum is conserved, so products recoil appropriately

For many H2 equation-balancing questions, the first two checks are the most operational.

Alpha Decay

An alpha particle is:

$${}_2^4\mathrm{He}$$

General form:

$${}_Z^{A}X\to {}_{Z-2}^{A-4}Y+{}_2^4\mathrm{He}$$

Changes:

• $A$ decreases by 4
• $Z$ decreases by 2

Example:

$${}_{92}^{238}\mathrm{U}\to {}_{90}^{234}\mathrm{Th}+{}_2^4\mathrm{He}$$

Check:

$$238=234+4$$ $$92=90+2$$

Beta-Minus Decay

In beta-minus decay, a neutron in the nucleus changes into a proton.

At nuclide level:

$${}_Z^{A}X\to {}_{Z+1}^{A}Y+{}_{-1}^{0}e$$

Changes:

• $A$ unchanged
• nuclear $Z$ increases by 1
• emitted beta-minus particle has charge number $-1$

Observed beta particles have a continuous range of energies. The daughter nucleus and beta particle alone cannot complete the observed energy and momentum bookkeeping, so an additional neutral particle—called the neutrino in the syllabus—was predicted. Detailed antineutrino notation is enrichment and is not required in a 9749 core answer.

Enrichment — Beyond H2 Physics 9749: Beta-Plus Decay

Positron emission is not required

This section is retained for modern-physics context. Do not treat its particle or antiparticle notation as assessed 9749 recall.

In beta-plus decay, a proton in the nucleus changes into a neutron.

At nuclide level:

$${}_Z^{A}X\to {}_{Z-1}^{A}Y+{}_{+1}^{0}e$$

Changes:

• $A$ unchanged
• nuclear $Z$ decreases by 1
• emitted beta-plus particle has charge number $+1$

Particle-level form:

$$p\to n+e^{+}+{\nu }_e$$

See Beta-Decay Particle Context.

Gamma Emission

Gamma emission is the release of electromagnetic radiation from an excited nucleus.

At nuclide level:

$${}_Z^A{X}^{\ast }\to {}_Z^{A}X+\gamma$$

Changes:

• $A$ unchanged
• $Z$ unchanged
• nuclear energy decreases

Fission and Fusion

In fission and fusion, nucleon number and charge are still conserved.

Energy release is explained by mass-energy conservation:

$$E=\Delta m{c}^2$$

The total rest mass of products can be less than the total rest mass of reactants. The missing rest energy appears as kinetic energy of products and/or radiation.

Useful wording:

• rest mass is not separately conserved
• total mass-energy is conserved
• products are more tightly bound when total binding energy increases

See Nuclear Fission and Nuclear Fusion.

Short Worked Examples

Example 1: Alpha Decay Daughter

Complete:

$${}_{88}^{226}\mathrm{Ra}\to ?+{}_2^4\mathrm{He}$$

Balance $A$:

$$226-4=222$$

Balance $Z$:

$$88-2=86$$

So:

$${}_{88}^{226}\mathrm{Ra}\to {}_{86}^{222}\mathrm{Rn}+{}_2^4\mathrm{He}$$

Example 2: Beta-Minus Daughter

For beta-minus decay:

$${}_6^{14}\mathrm{C}\to ?+{}_{-1}^{0}e$$

Balance $A$:

$$14=14+0$$

Balance $Z$:

$$6=7+(-1)$$

So:

$${}_6^{14}\mathrm{C}\to {}_7^{14}\mathrm{N}+{}_{-1}^{0}e$$

Example 3: Simple Induced Reaction

Complete

$${}_7^{14}\mathrm{N}+{}_2^4\mathrm{He}\to {}_8^{17}\mathrm{O}+X.$$

Balance the top numbers:

$$A_X=(14+4)-17=1.$$

Balance the bottom numbers:

$$Z_X=(7+2)-8=1.$$

Therefore

$$X={}_1^1\mathrm{H}.$$

Common Exam Traps

• Balancing $A$ but forgetting to balance $Z$.
• Treating beta-minus emission as loss of an orbital electron.
• Treating beta-plus emission as required 9749 core content.
• Saying gamma emission changes the element.
• Saying mass is conserved in fission or fusion without mentioning mass-energy.
• Stating that the daughter nucleus and beta particle alone account for all observed beta-decay energy and momentum.

Formula Summary

Idea	Representation	Use
Nuclide notation	${}_Z^{A}X$	Identify nucleon number and proton number.
Neutron number	$N=A-Z$	Find neutrons in a nucleus.
Alpha particle	${}_2^4\mathrm{He}$	Balance alpha decay.
Beta-minus particle	${}_{-1}^{0}e$	Balance beta-minus decay.
Enrichment: beta-plus particle	${}_{+1}^{0}e$	Read a supplied beta-plus equation only; not required recall.
Mass-energy	$E=\Delta m{c}^2$	Explain energy release.

Summary

Nuclear equations are governed by conservation laws. In most H2 questions, balance nucleon number and charge number first, then interpret the process using total mass-energy and momentum conservation. In beta decay, use the syllabus’s generic neutrino language unless extra notation is supplied.

Links

• Nuclear Physics
• Radioactive Decay
• Beta-Decay Particle Context
• Nuclear Fission
• Nuclear Fusion
• Conservation Laws in Physics