Chapter 14, Problem 14.33PAE

Interpretation Introduction

Interpretation: To write an equation for each step in this series.

${\text{α, β}}^{\text{-}}{\text{,β}}^{\text{-}}{\text{,α, α, α, α, β}}^{\text{-}}{\text{,β}}^{\text{-}}\text{,α}$

Concept introduction:

Isotopes:

These are the atoms of the similar element which contains the different number of neutrons and same number of protons.

For example:

$\begin{array}{l}\text{Isotopes of Carbon}\\ {}_{\text{6}}^{\text{12}}{\text{C }}_{\text{6}}^{\text{13}}{\text{C }}_{\text{ 6}}^{\text{ 14}}\text{C}\end{array}$

Isotopes of atoms with unstable nuclei are called radioisotopes.

Radioactive decay: Unstable isotopes release particles and produce energy to become stable isotopes.

An element undergoes radioactive decay process it changes into a different isotope of the similar element or into an entirely different element after radioactive decay.

Alpha decay ( $\text{α}$): Nucleus emits an alpha particle containing two protons and two neutrons (helium nucleus. Transformation occurs as a nucleus has four less particles.\

For example:

${}_{\text{64}}^{\text{149}}\text{Gd }\to {}_{\text{62}}^{\text{145}}{\text{Sm + }}_{\text{2}}^{\text{4}}\text{He}$

Beta Decay ( $\text{β}$):

A state where the nucleus of an atom has too many neutrons. The neutron transforms into a proton and an electron is released from the nucleus.

${}_{\text{6}}^{\text{14}}\text{C }\to {}_{\text{7}}^{\text{14}}{\text{N + }}_{\text{-1}}^{\text{0}}\text{e}$

Two types of $\text{β}$ decays occur.

In beta −minus decay, a neutron first breaks down into an electron and proton. After that, the electron is emitted from the nucleus, whereas the proton stays inside the nucleus. The resulting nucleus has one more proton and one less neutron.

In beta −plus decay, a proton first breaks down into a neutron and a positron. After that, the positron is emitted from the nucleus and the neutron stays inside the nucleus, the resulting nucleus has one less proton and one more neutron.

Answer to Problem 14.33PAE

Solution: The sequence of ${\text{α, β}}^{\text{-}}{\text{,β}}^{\text{-}}{\text{,α, α, α, α, β}}^{\text{-}}{\text{,β}}^{\text{-}}\text{,α}$ radioactive decay of ${}^{\text{232}}\text{Th}$ is as follows.

$\begin{array}{l}\text{α:}\\ {}_{\text{90}}^{\text{232}}\text{Th }\to {}_{\text{88}}^{\text{228}}{\text{Ra+}}_{\text{2}}^{\text{4}}\text{He}\\ {\text{β}}^{\text{-}}\text{:}\\ {}_{\text{88}}^{\text{228}}\text{Ra }{\to }_{\text{89}}^{\text{228}}{\text{Ac+}}_{\text{-1}}^{\text{0}}\text{β+}\overline{\text{ν}}\\ {\text{β}}^{\text{-}}\text{:}\\ {}_{\text{89}}^{\text{238}}\text{Ac }{\to }_{\text{90}}^{\text{228}}{\text{Th+}}_{\text{-1}}^{\text{0}}\text{β+}\overline{\text{ν}}\\ \text{α:}\\ {}_{\text{90}}^{\text{228}}\text{Th }{\to }_{\text{88}}^{\text{224}}{\text{Ra+}}_{\text{2}}^{\text{4}}\text{He}\\ \text{α:}\\ {}_{\text{88}}^{\text{224}}\text{Ra }{\to }_{\text{86}}^{\text{220}}{\text{Rn+}}_{\text{2}}^{\text{4}}\text{He}\\ \text{α:}\\ {}_{\text{86}}^{\text{220}}\text{Rn }{\to }_{\text{84}}^{\text{216}}{\text{Po+}}_{\text{2}}^{\text{4}}\text{He}\\ \text{α:}\\ {}_{\text{84}}^{\text{216}}\text{Po }{\to }_{\text{82}}^{\text{212}}{\text{Pb+}}_{\text{2}}^{\text{4}}\text{He}\\ {\text{β}}^{\text{-}}\text{:}\\ {}_{\text{82}}^{\text{212}}\text{Pb }{\to }_{\text{83}}^{\text{212}}{\text{Bi+}}_{\text{-1}}^{\text{0}}\text{β+}\overline{\text{ν}}\\ {\text{β}}^{\text{-}}\text{:}\\ {}_{\text{83}}^{\text{212}}\text{Bi }{\to }_{\text{84}}^{\text{212}}{\text{Po+}}_{\text{-1}}^{\text{0}}\text{β+}\overline{\text{ν}}\\ \text{α:}\\ {}_{\text{84}}^{\text{212}}\text{Po }{\to }_{\text{82}}^{\text{208}}{\text{Pb+}}_{\text{2}}^{\text{4}}\text{He}\end{array}$

Explanation of Solution

The ${}^{\text{232}}\text{Th}$ radioactive series beginning with ${}^{\text{232}}\text{Th}$ and ending with ${}^{\text{208}}\text{Pb}$.

The sequence occurs as follows.

${\text{α, β}}^{\text{-}}{\text{,β}}^{\text{-}}{\text{,α, α, α, α, β}}^{\text{-}}{\text{,β}}^{\text{-}}\text{,α}$

The radioactive decay series is as follows.

$\begin{array}{l}\text{α:}\\ {}_{\text{90}}^{\text{232}}\text{Th }\to {}_{\text{88}}^{\text{228}}{\text{Ra+}}_{\text{2}}^{\text{4}}\text{He}\\ {\text{β}}^{\text{-}}\text{:}\\ {}_{\text{88}}^{\text{228}}\text{Ra }{\to }_{\text{89}}^{\text{228}}{\text{Ac+}}_{\text{-1}}^{\text{0}}\text{β+}\overline{\text{ν}}\\ {\text{β}}^{\text{-}}\text{:}\\ {}_{\text{89}}^{\text{238}}\text{Ac }{\to }_{\text{90}}^{\text{228}}{\text{Th+}}_{\text{-1}}^{\text{0}}\text{β+}\overline{\text{ν}}\\ \text{α:}\\ {}_{\text{90}}^{\text{228}}\text{Th }{\to }_{\text{88}}^{\text{224}}{\text{Ra+}}_{\text{2}}^{\text{4}}\text{He}\\ \text{α:}\\ {}_{\text{88}}^{\text{224}}\text{Ra }{\to }_{\text{86}}^{\text{220}}{\text{Rn+}}_{\text{2}}^{\text{4}}\text{He}\\ \text{α:}\\ {}_{\text{86}}^{\text{220}}\text{Rn }{\to }_{\text{84}}^{\text{216}}{\text{Po+}}_{\text{2}}^{\text{4}}\text{He}\\ \text{α:}\\ {}_{\text{84}}^{\text{216}}\text{Po }{\to }_{\text{82}}^{\text{212}}{\text{Pb+}}_{\text{2}}^{\text{4}}\text{He}\\ {\text{β}}^{\text{-}}\text{:}\\ {}_{\text{82}}^{\text{212}}\text{Pb }{\to }_{\text{83}}^{\text{212}}{\text{Bi+}}_{\text{-1}}^{\text{0}}\text{β+}\overline{\text{ν}}\\ {\text{β}}^{\text{-}}\text{:}\\ {}_{\text{83}}^{\text{212}}\text{Bi }{\to }_{\text{84}}^{\text{212}}{\text{Po+}}_{\text{-1}}^{\text{0}}\text{β+}\overline{\text{ν}}\\ \text{α:}\\ {}_{\text{84}}^{\text{212}}\text{Po }{\to }_{\text{82}}^{\text{208}}{\text{Pb+}}_{\text{2}}^{\text{4}}\text{He}\end{array}$

Conclusion

The ${}^{\text{232}}\text{Th}$ radioactive series beginning with ${}^{\text{232}}\text{Th}$ and ending with ${}^{\text{208}}\text{Pb}$.