Nuclear Equations and Conservation

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.
• Beta-plus decay keeps $A$ unchanged and decreases nuclear $Z$ by 1.
• 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 missing particles such as neutrinos are included in fuller beta-decay descriptions and why fission or fusion can release energy.

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 $-1$

Particle-level form:

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

The antineutrino helps account for energy and momentum in the fuller description.

Beta-Plus Decay

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:

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

Common Exam Traps

• Balancing $A$ but forgetting to balance $Z$.
• Treating beta-minus emission as loss of an orbital electron.
• Forgetting that beta-plus emission decreases nuclear $Z$.
• Saying gamma emission changes the element.
• Saying mass is conserved in fission or fusion without mentioning mass-energy.
• Ignoring neutrinos in fuller beta-decay particle equations when they are required.

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.
Beta-plus particle	${}_{+1}^{0}e$	Balance beta-plus decay.
Mass-energy	$E=\Delta m{c}^2$	Explain energy release.

Summary

Nuclear equations are governed by conservation laws. In most H2 questions, the fastest route is to balance nucleon number and charge number first, then interpret the physical process using decay type, particle context, and mass-energy conservation.

Links

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