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Chapter 24: Problem 5

In a radioactive decay chain, \({ }_{90}^{232} \mathrm{Th}\) nucleus decays to \({ }_{82}^{212} \mathrm{~Pb}\) nucleus. Let \(N_{\alpha}\) and \(N_{\beta}\) be the number of \(\alpha\) and \(\beta^{-}\) particles, respectively, emitted in this decay process. Which of the following statements is (are) true? (A) \(N_{\alpha}=5\) (B) \(N_{\alpha}=6\) (C) \(N_{\beta}=2\) (D) \(N_{\beta}=4\)

Short Answer

Expert verified
N_{\beta}=0, which make the correct statements (A), (C)

Step by step solution

01

Analyze Alpha Decay

An alpha decay (- particles emitted. Since an alpha particle is , each alpha decay will reduce the mass number by 4 and the atomic number by 2.
02

Calculate Number of Alpha Particles

The initial nucleus is and the final nucleus is . The difference in atomic number is is 90 - 82 = 8. It takes inner 'content' field
03

Analyze Beta Decay

A beta decay (), emitting an electron and an antineutrino. For each beta decay, the atomic number increases by 1, but the mass number remains unchanged.
04

Calculate Number of Beta Particles

After the alpha decays, the atomic number must reach 82, the atomic number of Lead (. Knowing that alpha decays have already reduced the atomic number to 82, any additional increase must be due to beta decays. Thus, we must determine how many beta decays occurred to raise the atomic number to the level of the final product without affecting the mass number.
05

Determine True Statements

The number of alpha particles emitted, , and the number of beta particles emitted, . From the analysis, and . Thus, the true statements are the ones that match our calculated numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Alpha Decay
Alpha decay is a type of radioactive decay wherein an unstable nucleus emits an alpha particle, essentially a helium-4 nucleus, to become a more stable element. This process results in the reduction of the mass number (A) of the parent nuclide by four units and the atomic number (Z) by two units.

For example, when \( { }_{90}^{232} \mathrm{Th} \) undergoes alpha decay, it ejects an alpha particle (\( { }_{2}^{4} \mathrm{He} \) nucleus) and transforms into a new nucleus with a mass number of 228 and an atomic number of 88. The equation for this decay process is: \[ { }_{90}^{232} \mathrm{Th} \rightarrow { }_{88}^{228} \mathrm{Ra} + { }_{2}^{4} \mathrm{He} .\]

In the context of JEE Advanced Physics, understanding alpha decay is crucial, as it forms a part of the nuclear physics curriculum and is often tested in this prestigious examination.
Beta Decay
Beta decay is another form of radioactive decay, in which a nucleus emits a beta particle, which can be either a high-speed electron or positron. During \(\beta^{-}\) decay, a neutron transforms into a proton, an electron, and an antineutrino. The emitted electron is called the beta particle. This decay results in an increase in the atomic number by one, while the mass number remains unchanged.

The decay of \( { }_{90}^{232} \mathrm{Th} \) involves both alpha and beta decays to eventually form \( { }_{82}^{212} \mathrm{~Pb} \). Since the mass number doesn't change during beta decay, it's the alpha decays that account for the mass number change from 232 to 212, but beta decays increase the atomic number from the intermediate product to lead's atomic number. This is key to solving problems like the one under consideration and is often tested in competitive exams like JEE Advanced, making it a vital concept for nuclear physics topics.
Nuclear Physics
Nuclear physics is a branch of physics that deals with the constituents and interactions of atomic nuclei. Core topics include the study of radioactivity, fusion, fission, and the behavior of nuclei under various conditions. Principles of nuclear physics are applied in various fields such as medical imaging, power generation, and understanding cosmological phenomena.

Within the scope of JEE Advanced Physics, students must comprehend how fundamental interactions manifest in phenomena like radioactive decay chains. For instance, they need to solve problems based on the sequential transformation of nuclei involving alpha and beta decays, requiring a clear grasp of how mass numbers and atomic numbers change during these processes. As these topics are complex and require high analytical skills, they are perfect candidates for challenging questions in competitive exams like JEE Advanced.
JEE Advanced Physics
The Joint Entrance Examination (JEE) Advanced Physics section assesses a student’s understanding of fundamental physics concepts at an advanced level. It is known for its challenging problems that test application, analytical thinking, and conceptual clarity. Radioactive decay chains, involving alpha and beta particles, regularly feature in the exam.

Answering these types of questions correctly requires the application of nuclear physics principles. As demonstrated in the original exercise, knowing the atomic number and mass number adjustments for each decay type is crucial for deducing the number of alpha and beta particles emitted during a decay sequence. Being adept in such problems facilitates success in JEE Advanced Physics and is critical for aspiring engineers and scientists who wish to enter premier institutes in India.

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Most popular questions from this chapter

Consider an electrochemical cell: \(\mathrm{A}(\mathrm{s})\left|\mathrm{A}^{\mathrm{n}+}(\mathrm{aq}, 2 \mathrm{M}) \| \mathrm{B}^{2 \mathrm{n}+}(\mathrm{aq}, 1 \mathrm{M})\right| \mathrm{B}(\mathrm{s})\). The value of \(\Delta H^{\ominus}\) for the cell reaction is twice that of \(\Delta G^{\theta}\) at \(300 \mathrm{~K}\). If the emf of the cell is zero, the \(\Delta S^{\theta}\) (in \(\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\) ) of the cell reaction per mole of \(\mathrm{B}\) formed at \(300 \mathrm{~K}\) is (Given: \(\ln (2)=0.7, R\) (universal gas constant) \(=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} \cdot H, S\) and \(G\) are enthalpy, entropy and Gibbs energy, respectively.)

In the List-I below, four different paths of a particle are given as functions of time. In these functions, \(\alpha\) and \(\beta\) are positive constants of appropriate dimensions and \(\alpha \neq \beta .\) In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: \(\vec{p}\) is the linear momentum, \(\vec{L}\) is the angular momentum about the origin, \(K\) is the kinetic energy, \(U\) is the potential energy and \(E\) is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path. LIST-I P. \(\vec{r}(t)=\alpha t \hat{\imath}+\beta t \hat{\jmath}\) Q. \(\vec{r}(t)=\alpha \cos \omega t \hat{\imath}+\beta \sin \omega t \hat{\jmath}\) R. \(\vec{r}(t)=\alpha(\cos \omega t \hat{\imath}+\sin \omega t \hat{\jmath})\) S. \(\vec{r}(t)=\alpha t \hat{\imath}+\frac{\beta}{2} t^{2} \hat{\jmath}\) LIST-II 1\. \(\vec{p}\) 2\. \(\vec{L}\) 3\. \(K\) 4\. \(U\) 5\. \(E\)

A planet of mass \(M\), has two natural satellites with masses \(m_{1}\) and \(m_{2}\). The radii of their circular orbits are \(R_{1}\) and \(R_{2}\) respectively. Ignore the gravitational force between the satellites. Define \(v_{1}, L_{1}, K_{1}\) and \(T_{1}\) to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1 ; and \(v_{2}, L_{2}, K_{2}\) and \(T_{2}\) to be the corresponding quantities of satellite 2. Given \(m_{1} / m_{2}=2\) and \(R_{1} / R_{2}=1 / 4\), match the ratios in List-I to the numbers in List-II. LIST-I P. \(\frac{v_{1}}{v_{2}}\) Q. \(\frac{L_{1}}{L_{2}}\) \(\mathbf{R} \quad \frac{K_{1}}{K_{2}}\) S. \(\frac{T_{1}}{T_{2}}\) LIST-II 1\. \(\frac{1}{8}\) 2\. 1 3\. 2 4\. 8

In a high school, a committee has to be formed from a group of 6 boys \(M_{1}, M_{2}, M_{3}, M_{4}, M_{5}, M_{6}\) and 5 girls \(G_{1}, G_{2}, G_{3}, G_{4}, G_{5}\) (i) Let \(\alpha_{1}\) be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls. (ii) Let \(\alpha_{2}\) be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls. (iii) Let \(\alpha_{3}\) be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls. (iv) Let \(\alpha_{4}\) be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls and such that both \(M_{1}\) and \(G_{1}\) are NOT in the committee together. LIST-I P. The value of \(\alpha_{1}\) is Q. The value of \(\alpha_{2}\) is \(\mathbf{R}\). The value of \(\alpha_{3}\) is \(\mathrm{S}\). The value of \(\alpha_{4}\) is LIST-II 1\. 136 2\. 189 3\. 192 4\. 200 5\. 381 6\. 461

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass \(m=0.4 \mathrm{~kg}\) is at rest on this surface. An impulse of \(1.0 \mathrm{~N} \mathrm{~s}\) is applied to the block at time \(t=0\) so that it starts moving along the \(x\) -axis with a velocity \(v(t)=v_{0} e^{-t / \tau}\), where \(v_{0}\) is a constant and \(\tau=4 \mathrm{~s}\). The displacement of the block, in metres, at \(t=\tau\) is Take \(e^{-1}=0.37\).
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