# 8.3.2 Decay Equations

### Changes in N and Z by Radioactive Decay

• There are four reasons why a nucleus might become unstable, and these determine which decay mode will occur

• Too many neutrons
• Decays through beta-minus (β) emission
• One of the neutrons in the nucleus changes into a proton and a β particle (an electron) and antineutrino is released
• The nucleon number is constant, while the proton number increases by 1
• The general decay equation for β emission is:

• Too many protons
• Decays through beta-plus (β+) emission or electron capture
• In beta-plus decay, a proton changes into a neutron and a β+ particle (a positron) and neutrino are released
• In electron capture, an orbiting electron is taken in by the nucleus and combined with a proton causing the formation of a neutron and neutrino
• In both types of decay, the nucleon number stays constant, while the proton number decreases by 1
• The general decay equation for β+ emission is:

• The equation for electron capture is:

• Too many nucleons
• Decays through alpha (α) emission
• An α particle is a helium nucleus
• The nucleon number decreases by 4 and the proton number decreases by 2
• The general decay equation for α emission is:

• Too much energy
• Decays through gamma (γ) emission
• A gamma particle is a high-energy electromagnetic radiation
• This usually occurs after a different type of decay, such as alpha or beta decay
• This is because the nucleus becomes excited and has excess energy
• In summary, alpha decay, beta decay and electron capture can be represented on an N–Z graph as follows:

#### Worked Example

Plutonium-239 is a radioactive isotope that contains 94 protons and emits α particles to form a radioactive isotope of uranium. This isotope of uranium emits α particles to form an isotope of thorium which is also radioactive.

a) Write two equations to represent the decay of plutonium-239 and the subsequent decay of uranium

b) Predict the decay mode of the thorium isotope

c) Draw the decay chain from plutonium-239 to the daughter product of thorium decay on an N–Z graph

Part (a)

Step 1: Write down the general equation of alpha decay

Step 2: Write down the decay equation of plutonium into uranium

Step 3: Write down the decay equation of uranium into thorium

Part (b)

• Plutonium, 239Pu
• Number of neutrons: 239 – 94 = 145
• Neutron-nucleon ratio: 145 / 239 = 0.607
• Uranium, 235U
• Number of neutrons: 235 – 92 = 143
• Neutron-nucleon ratio: 143 / 235 = 0.609
• Thorium, 231Th
• Number of neutrons: 231 – 90 = 141
• Neutron-nucleon ratio: 141 / 231 = 0.610
• Thorium-231 is neutron-rich compared to uranium-235 and plutonium-239
• Therefore, it must be a β emitter

Part (c)

• The key features to draw on an N–Z graph are:
• Values for neutron number (N) on the vertical axis
• Values for proton number (Z) on the horizontal axis
• Labels for the isotopes eg. 239Pu, 235U, 231Th
• Arrows showing the direction of the decay
• Labels for the type of emission eg. α, β
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