Problem 11 - College Physics

Question

Thorium series. The following decays make up the thorium decay series (the \(X\) 's are unknowns for you to identify): $$^{232} \mathrm{Th} \stackrel{\alpha}{\longrightarrow} X_{1}, \quad^{228} \mathrm{Ra} \stackrel{\beta^{-}}{\longrightarrow}^{228} \mathrm{Ac}, \quad X_{2} \stackrel{\beta^{-}}{\longrightarrow}^{228} \mathrm{Th}$$ $$^{228} \mathrm{Th} \stackrel{x_{3}}{\longrightarrow}^{224} \mathrm{Ra}, \quad^{224} \mathrm{Ra} \stackrel{\alpha}{\longrightarrow}^{220} \mathrm{Rn}, \quad^{220} \mathrm{Rn} \stackrel{\alpha}{\longrightarrow} X_{4}$$ \(X_{5} \stackrel{\alpha}{\longrightarrow}^{212} \mathrm{Pb},\) and \(^{212 \mathrm{Pb}} \stackrel{x_{6}}{\longrightarrow}^{212} \mathrm{Bi} .\) The \(^{212} \mathrm{Bi}\) then decays by an \(\alpha\) decay and a \(\beta^{-}\) decay, which can occur in either order \((\alpha\) followed by \(\beta\) or \(\beta\) followed by \(\alpha)\) . (a) Identify each of the six unknowns \((X_{1}, X_{2},\) etc. \()\) by nucleon number, atomic number, neutron number, and name. (b) Write out the decays of \(^{212} \mathrm{Bi}\) and indicate the end product of this series. (For some guidance, see the discussion under "Decay Series" in Section \(30.3 . )(\mathrm{c})\) Draw a Segre chart for the thorium series, similar to the one shown in Figure \(30.5 .\)

Step-by-Step Solution

Answer

(a) Identify the isotopes: X_1= ^{228}Ra, X_2= ^{228}Ac, X_4= ^{216}Po, X_5= ^{212}Pb. (b) End product is ^{208}Pb. (c) A Segre chart maps these decay steps.

Step 1: Identify X_1 in the Decay Sequence

The decay sequence starts with \(^{232} \mathrm{Th} \rightarrow X_1\) via \(\alpha\) decay. An \(\alpha\) particle consists of 2 protons and 2 neutrons, reducing the atomic number by 2 and the mass number by 4.Thus, \(X_1\) is \(^{228}_{88}Rn\) (Radon is incorrect here, it should be \(Ra\) for radium.)

Step 2: Identify X_2 in the Decay Sequence

The sequence \(X_2 \rightarrow \ ^{228} \mathrm{Th}\) involves \(\beta^{-}\) decay, which increases the atomic number by 1.Thus, \(X_2\) must be \(^{228}_{89}Ac\) (Actinium).

Step 3: Identify X_3 Decay Type

Given \(^{228} \mathrm{Th} \rightarrow \ ^{224} \mathrm{Ra}\), this transition is an \(\alpha\) decay reducing the atomic number by 2 and the mass number by 4.

Step 4: Identify X_4 in the Decay Sequence

The sequence \(^{220} \mathrm{Rn} \rightarrow X_4\) involves \(\alpha\) decay.Thus, \(X_4\) is \(^{216}_{84}Po\) (Polonium).

Step 5: Identify X_5 in the Decay Sequence

The sequence \(X_5 \rightarrow \ ^{212} \mathrm{Pb}\) involves \(\alpha\) decay, reducing both atomic and mass numbers.Thus, \(X_5\) is \(^{216}_{86}Pb\) (Lead).

Step 6: Identify Type of Decay for X_6

Given the equation \(^{212} \mathrm{Pb} \rightarrow ^{212} \mathrm{Bi}\), it follows a \(\beta^-\) decay, increasing the atomic number by 1.

Step 7: Write Decay of ^{212}Bi and End Products

\(^{212} \mathrm{Bi}\rightarrow \ {}\begin{{cases}} ^{208}_{81}Tl \^{208}_{82}Pb \end{{cases}} \rightarrow ^{208}_{82}Pb\)This mixture of alpha and beta decay results in stable \(Pb\), with atomic number 82 and mass number 208.

Step 8: Drawing the Segre Chart

The Segre chart is a graph where the y-axis represents the neutron number and the x-axis represents the proton (atomic) number. Plot the decay sequence showing paths from Th-232 down to stable Pb-208.1. Plot initial \(^{232}Th (Z=90, N=142)\).2. Represent each decay action path with downward lines reducing protons and mass numbers consistent with decay particles released.Ultimately, plot the sequence leading to \(^{208}Pb (Z=82, N=126)\).

Key Concepts

Alpha Decay

Alpha decay is a type of radioactive decay where an alpha particle, made up of 2 protons and 2 neutrons, is emitted from a nucleus. This process decreases the atomic number by 2 and the mass number by 4. As a result, the element transforms into a new element that is two places back on the periodic table.
In the thorium decay series, several steps involve alpha decay, such as the transition from \[^{232} \text{Th} \rightarrow ^{228} \text{Ra}\] and \[^{224} \text{Ra} \rightarrow ^{220} \text{Rn}\]. To identify the new element formed, you can think of it like shedding small parts of the nucleus, leaving you with a slightly lighter and less positively charged core. The beauty of alpha decay lies in its predictability as the nucleus steadily shifts backwards to a stable state, like our end product of lead (Pb).

Beta Decay

Beta decay involves the transformation of a neutron into a proton, with the emission of an electron (beta particle) and an antineutrino. This increases the atomic number by 1 while leaving the mass number unchanged. It allows the element to jump forward by one place on the periodic table.
Within the thorium decay series, beta decay contributes to transitions such as \[^{228} \text{Ra} \rightarrow ^{228} \text{Ac}\]and \[^{212} \text{Pb} \rightarrow ^{212} \text{Bi}\].In these processes, the elements are effectively increasing their proton count to become new elements. Think about beta decay as a domino effect, where a neutron's change directly impacts the identity of the atom, nudging it closer to stability.

Nuclear Chemistry

Nuclear chemistry involves the study of the changes in atomic nuclei. In particular, this field examines radioactive decay, nuclear properties, and nuclear reactions. The thorium decay series is an exemplary study in nuclear chemistry, demonstrating how a large unstable nucleus transforms into a more stable form through consecutive decay processes.
Understanding nuclear chemistry allows us to predict the behavior of unstable isotopes and to harness nuclear processes for various applications, such as energy generation. This part of chemistry uniquely combines aspects of physics and chemistry to explain phenomena at an atomic level, providing insights not only into decay processes but also into the forces that stabilize a nucleus.

Transmutation

Transmutation is the process of converting one chemical element or isotope into another, a central feature of radioactive decay like that seen in the thorium series. Through decay processes such as alpha and beta decay, the identity of an element changes as it loses part of its nucleus or changes its atomic structure.
For example, during alpha decay, the emission of an alpha particle results in a reduction in both atomic and mass numbers, effectively changing the element into another. Similarly, beta decay increases the atomic number, leading to a different element. Through transmutation, radioactive series gradually lead elements towards a stable form. Transmutation is fascinating because it blurs the rigid boundaries of the periodic table, allowing elements to shift identities and stabilize over time.

Radioactive Decay Series

A radioactive decay series is a sequence of decays that certain isotopes undergo until reaching a stable form. The thorium decay series is one example, starting from an unstable parent isotope, \[^{232} \text{Th}\], and moving through a series of alpha and beta decays to eventually form stable \[^{208} \text{Pb}\].
Throughout this series, isotopes systematically release particles, decreasing in energy and gradually transforming into different elements. Each decay chain offers a roadmap of the elemental transition, providing insights into nuclear stability and transformations over time. These series are not just theoretical constructs but also guide us in real-life applications such as nuclear medicine, radiometric dating, and understanding natural radioactive decay occurring on Earth.