Question

The parent nuclide of the thorium decay series is 232 90 \(\mathrm{Th}\) The first four decays are as follows: alpha emission, beta emission, beta emission, and alpha emission. Write the nuclear equations for this series of emissions.

Short answer

The equations are \(^{232}_{90} \mathrm{Th} \rightarrow ^{228}_{88} \mathrm{Ra} + \ ^{4}_{2} \mathrm{He}\), \(^{228}_{88} \mathrm{Ra} \rightarrow^{228}_{89} \mathrm{Ac} + \ ^{0}_{-1} \beta\), \(^{228}_{89} \mathrm{Ac} \rightarrow^{228}_{90} \mathrm{Th} + \ ^{0}_{-1} \beta\), \(^{228}_{90} \mathrm{Th} \rightarrow ^{224}_{88} \mathrm{Ra} + \ ^{4}_{2} \mathrm{He}\)

Step-by-step solution

- Write the initial nuclide

The initial nuclide in the thorium decay series is given as \(^{232}_{90} \mathrm{Th}\).

- Alpha emission

When \(^{232}_{90} \mathrm{Th}\) undergoes alpha emission, it loses an alpha particle \(^{4}_{2}\mathrm{He}\). This transforms into a new element:\[^{232}_{90} \mathrm{Th} \rightarrow \ ^{228}_{88} \mathrm{Ra} + \ ^{4}_{2} \mathrm{He}\]

- First beta emission

The product \(^{228}_{88} \mathrm{Ra}\) now undergoes a beta emission, resulting in the emission of a beta particle \(^{0}_{-1}\beta\). This changes into another element:\[^{228}_{88} \mathrm{Ra} \rightarrow\ ^{228}_{89} \mathrm{Ac} + \ ^{0}_{-1} \beta\]

- Second beta emission

Next, \(^{228}_{89} \mathrm{Ac}\) undergoes another beta emission, releasing another \(^{0}_{-1}\beta\). The resulting transformation is:\[^{228}_{89} \mathrm{Ac} \rightarrow\ ^{228}_{90} \mathrm{Th} + \ ^{0}_{-1} \beta\]

- Second alpha emission

Finally, \(^{228}_{90} \mathrm{Th}\) undergoes alpha emission, releasing an \(^{4}_{2}\mathrm{He}\). This further decays into:\[^{228}_{90} \mathrm{Th} \rightarrow ^{224}_{88} \mathrm{Ra} + \ ^{4}_{2} \mathrm{He}\]

Key concepts

alpha emission

Thorium-232, written as \(^{232}_{90} \mathrm{Th}\), undergoes a type of radioactive decay known as alpha emission. In alpha emission, an alpha particle, which consists of 2 protons and 2 neutrons (essentially a helium-4 nucleus \(^{4}_{2}\mathrm{He}\)), is emitted from the nucleus. This results in the loss of 4 mass units and 2 protons from the original nucleus.
Here’s what happens mathematically:
\[^{232}_{90} \mathrm{Th} \rightarrow \, ^{228}_{88} \mathrm{Ra} + \, ^{4}_{2} \mathrm{He}\]
This changes the element Thorium (Th) into Radium (Ra), while emitting an alpha particle.

beta emission

Beta emission is another common type of radioactive decay, but it works differently from alpha emission. During beta emission, a neutron in the nucleus is converted into a proton, and a beta particle (essentially an electron \(^{0}_{-1} \beta\)) is emitted. This increases the atomic number by 1 while keeping the mass number the same.
Let’s break this down with our example:

• First Beta Emission:
  \[^{228}_{88} \mathrm{Ra} \rightarrow \, ^{228}_{89} \mathrm{Ac} + \, ^{0}_{-1} \beta\]
• Second Beta Emission:
  \[^{228}_{89} \mathrm{Ac} \rightarrow \, ^{228}_{90} \mathrm{Th} + \, ^{0}_{-1} \beta\]

After two beta emissions, the nucleus takes on different elemental forms: Radium (Ra) changes to Actinium (Ac), and then Actinium further changes into Thorium (Th).

nuclear equations

Nuclear equations are used to represent the changes occurring in the nucleus during radioactive decay. These equations must balance both the atomic numbers and the mass numbers on both sides. It’s essential to correctly sum the protons and neutrons (atomic and mass numbers) to reflect the conservation laws in nuclear reactions.
In the example we explored, we start with Thorium-232 which undergoes several transformations:

• Alpha Emission:
  \[^{232}_{90} \mathrm{Th} \rightarrow ^{228}_{88} \mathrm{Ra} + ^{4}_{2} \mathrm{He}\]
• First Beta Emission:
  \[^{228}_{88} \mathrm{Ra} \rightarrow ^{228}_{89} \mathrm{Ac} + ^{0}_{-1} \beta\]
• Second Beta Emission:
  \[^{228}_{89} \mathrm{Ac} \rightarrow ^{228}_{90} \mathrm{Th} + ^{0}_{-1} \beta\]
• Second Alpha Emission:
  \[^{228}_{90} \mathrm{Th} \rightarrow ^{224}_{88} \mathrm{Ra} + ^{4}_{2} \mathrm{He}\]

Each step adheres to the principle of conservation of mass and atomic numbers, providing a clear picture of the radioactive decay process.