Decay Equations and Conservation

Overview

Decay Equations and Conservation focuses on writing and interpreting nuclear decay equations correctly.

This page deepens ideas from:

• Radioactive Decay
• Nuclear Physics
• Conservation Laws in Physics

To solve decay-equation questions, always apply:

• conservation of nucleon number
• conservation of charge, using proton number for nuclei and charge number for emitted particles

Definition

Decay equations represent nuclear transformations using nuclide notation and must satisfy the relevant conservation laws.

Why It Matters

This idea matters because students must be able to:

• identify the emitted radiation
• determine the daughter nucleus correctly
• check that an equation is physically consistent

Key Representations

Nuclide Notation

A nucleus is written as:

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

where:

• $A$ = nucleon number
• $Z$ = proton number for a nucleus
• $X$ = element symbol

Neutron number:

$$N=A-Z$$

Example:

$${}_{92}^{238}U$$

• protons = 92
• neutrons = 146

Core Conservation Rules

1. Nucleon Number Conserved

The total top numbers must balance.

2. Charge Conserved

The total bottom numbers must balance.

For nuclei, the bottom number is proton number. For beta particles, it represents charge number.

3. Energy Conserved

Energy may appear as:

• kinetic energy of emitted particles
• gamma radiation
• rest-mass change accounted for by mass-energy conservation

4. Momentum Conserved

Products recoil appropriately.

For H2 equation balancing, the first two are usually the most important.

Alpha Decay Equations

Particle emitted:

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

or

$$\alpha$$

General form:

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

Changes:

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

Example:

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

Check:

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

Beta-Minus Decay Equations

Particle emitted:

$${}_{-1}^{0}e$$

or

$${\beta }^{-}$$

A neutron in the nucleus becomes a proton.

General form:

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

The antineutrino may also be included.

Changes:

• $A$ unchanged
• $Z$ increases by 1

Example:

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

Check:

$$14=14+0$$ $$6=7+(-1)$$

Beta-Plus Decay Equations

Particle emitted:

$${}_{+1}^{0}e$$

or

$${\beta }^{+}$$

A proton in the nucleus becomes a neutron.

General form:

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

The neutrino may also be included.

Changes:

• $A$ unchanged
• $Z$ decreases by 1

Gamma Emission Equations

Radiation emitted:

$$\gamma$$

Gamma radiation is electromagnetic radiation from an excited nucleus.

General form:

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

Changes:

• $A$ unchanged
• $Z$ unchanged

Only the nuclear energy decreases.

Example:

$${}_{27}^{60}C{o}^{\ast }\to {}_{27}^{60}Co+\gamma$$

How A and Z Change Summary

Decay Type	Change in $A$	Change in $Z$
Alpha	-4	-2
Beta-minus	0	+1
Beta-plus	0	-1
Gamma	0	0

Identifying Daughter Nuclei

Step method:

Alpha Decay

• subtract 4 from $A$
• subtract 2 from $Z$

Beta-Minus Decay

• keep $A$
• add 1 to $Z$

Beta-Plus Decay

• keep $A$
• subtract 1 from $Z$

Gamma Emission

• no change

Then identify the new element using proton number.

Worked Examples

Example 1: Alpha Decay

$${}_{88}^{226}Ra\to ?$$

After alpha decay:

• $A=222$
• $Z=86$

Element with $Z=86$ is Rn.

Answer:

$${}_{86}^{222}Rn+{}_2^4\text{He}$$

Example 2: Beta-Minus Decay

$${}_{53}^{131}I\to ?$$

After beta-minus decay:

• $A=131$
• $Z=54$

Element with $Z=54$ is Xe.

Answer:

$${}_{54}^{131}Xe+{}_{-1}^{0}e$$

Example 3: Missing Particle

$${}_{84}^{210}Po\to {}_{82}^{206}Pb+?$$

Difference:

• top: 4
• bottom: 2

So the particle is:

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

Example 4: Two Successive Beta-Minus Decays

Starting from:

$${}_{38}^{90}Sr$$

After the first beta-minus decay:

$${}_{39}^{90}Y$$

After the second beta-minus decay:

$${}_{40}^{90}Zr$$

Summary

• nuclear equations conserve nucleon number and charge
• alpha decay emits ${}_2^4\text{He}$
• beta-minus decay emits ${}_{-1}^{0}e$
• beta-plus decay emits ${}_{+1}^{0}e$
• gamma emission changes energy only
• momentum and mass-energy are also conserved, though $A$ and $Z$ are usually the main equation-balancing checks
• changes in $A$ and $Z$ identify the daughter nucleus

Links

• Radioactive Decay
• Nuclear Physics
• Conservation Laws in Physics
• Particle Physics
• Half-Life
• Radioactive Decay Common Exam Traps