NCERT Solutions Class 12 Physics Chapter 13: Nuclei Important Questions

Introduction

Class 12 Physics is a critical juncture in a student's academic journey, particularly in understanding complex concepts in modern physics. Chapter 13 of the NCERT Physics textbook, "Nuclei," delves into the fascinating world of atomic nuclei, covering essential topics that include nuclear structure, binding energy, radioactivity, and nuclear reactions. This article aims to provide a comprehensive analysis of the important questions from this chapter, offering detailed explanations and solutions to help students grasp these concepts effectively.

1. Structure of the Nucleus

1.1 What is the nuclear model of an atom?

The nuclear model of an atom, proposed by Ernest Rutherford in 1911, posits that the atom consists of a small, dense nucleus surrounded by electrons. The nucleus, which contains protons and neutrons, accounts for most of the atom's mass. The electrons orbit the nucleus in defined energy levels. This model replaced the earlier plum pudding model and laid the foundation for further developments in atomic theory.

1.2 Explain the concept of nuclear force.

Nuclear force is the force that acts between the nucleons (protons and neutrons) within the nucleus. Unlike electromagnetic forces, which decrease rapidly with distance, nuclear forces are short-range and work effectively only at distances on the order of a femtometer (10^-15 meters). This force is attractive and helps to overcome the electrostatic repulsion between positively charged protons, thus stabilizing the nucleus.

1.3 Define binding energy. How is it related to the stability of a nucleus?

Binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons. It is a measure of the nucleus's stability; the higher the binding energy, the more stable the nucleus. Binding energy is calculated using the mass defect, which is the difference between the mass of the nucleus and the sum of the masses of its individual nucleons. A nucleus with higher binding energy per nucleon is more stable.

2. Radioactivity

2.1 What are the different types of radioactive decay?

Radioactive decay occurs in three main types:

• Alpha Decay: In alpha decay, an unstable nucleus emits an alpha particle (two protons and two neutrons). This results in a new nucleus with an atomic number decreased by two and a mass number decreased by four.

• Beta Decay: Beta decay involves the emission of a beta particle. There are two types: beta-minus (β-) and beta-plus (β+). In β- decay, a neutron transforms into a proton, emitting an electron and an antineutrino. In β+ decay, a proton transforms into a neutron, emitting a positron and a neutrino.

• Gamma Decay: Gamma decay occurs when an excited nucleus releases energy in the form of gamma radiation (high-energy photons) without changing its atomic number or mass number.

2.2 Explain the concept of half-life with an example.

Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is a measure of the rate of decay of a radioactive substance. For example, if a substance has a half-life of 5 years, after 5 years, half of the original amount of the substance will have decayed. This process continues, with the amount of the substance decreasing by half every 5 years.

2.3 How is the activity of a radioactive substance related to its decay constant?

The activity (A) of a radioactive substance is the rate at which decay events occur. It is related to the decay constant (λ) by the equation:

$A = λN$

where $N$ is the number of radioactive nuclei present. The decay constant is a measure of the probability of decay per unit time. A higher decay constant implies a higher rate of decay and thus a higher activity.

3. Nuclear Reactions

3.1 Describe the process of nuclear fission and its applications.

Nuclear fission is the process in which a heavy nucleus splits into two lighter nuclei, along with the release of a significant amount of energy. This reaction is initiated when a heavy nucleus absorbs a neutron. The most common application of nuclear fission is in nuclear power plants, where it is used to generate electricity. Additionally, nuclear fission is the principle behind atomic bombs.

3.2 What is nuclear fusion? How does it differ from fission?

Nuclear fusion is the process in which two light nuclei combine to form a heavier nucleus, releasing energy in the process. Unlike fission, which splits heavy nuclei, fusion combines light nuclei, such as hydrogen isotopes, to form helium. Fusion powers stars, including the sun, and has the potential for use in energy production on Earth, though it is technologically challenging to achieve and sustain.

3.3 Explain the concept of mass-energy equivalence in the context of nuclear reactions.

Mass-energy equivalence, expressed by Einstein's famous equation $E = mc^2$, states that mass can be converted into energy and vice versa. In nuclear reactions, the mass of the products is often less than the mass of the reactants. The "missing" mass has been converted into energy, which is released during the reaction. This principle underpins the immense energy released in both nuclear fission and fusion.

4. Nuclear Models and Theories

4.1 Describe the liquid drop model of the nucleus.

The liquid drop model, proposed by George Gamow and others, likens the nucleus to a drop of incompressible liquid. This model accounts for the binding energy of the nucleus by considering factors such as surface tension, Coulomb repulsion, and the volume of the nucleus. It provides an understanding of nuclear properties and helps explain phenomena like nuclear fission.

4.2 What is the shell model of the nucleus?

The shell model of the nucleus, developed by Maria Goeppert Mayer and J. Hans D. Jensen, describes nucleons as occupying discrete energy levels or "shells" within the nucleus, similar to electron shells in atoms. This model explains nuclear stability, magic numbers (numbers of protons or neutrons that result in particularly stable nuclei), and the distribution of nuclear energy levels.

4.3 Explain the concept of nuclear binding energy and its calculation.

Nuclear binding energy is the energy required to separate a nucleus into its constituent protons and neutrons. It is calculated using the mass defect method:

$\text{Binding Energy} = (\text{Mass of nucleons} - \text{Mass of nucleus}) \times c^2$

where $c$ is the speed of light. The mass defect is the difference between the total mass of the individual nucleons and the mass of the nucleus. This energy corresponds to the stability of the nucleus, with higher binding energies indicating greater stability.

5. Important Questions for Practice

5.1 Calculate the binding energy per nucleon for a nucleus with a mass number of 56 and an actual mass of 55.928 amu.

To calculate the binding energy per nucleon:

1. Calculate the mass defect:

$\text{Mass Defect} = \text{Mass of nucleons} - \text{Actual mass of nucleus}$

For a nucleus with mass number 56:

$\text{Mass of nucleons} = 56 \text{ amu}$

$\text{Mass Defect} = 56 - 55.928 = 0.072 \text{ amu}$

2. Convert mass defect to energy:

$\text{Binding Energy} = 0.072 \text{ amu} \times 931.5 \text{ MeV/amu} = 67.2 \text{ MeV}$

3. Calculate binding energy per nucleon:

$\text{Binding Energy per Nucleon} = \frac{67.2 \text{ MeV}}{56} \approx 1.2 \text{ MeV}$

5.2 Determine the activity of a radioactive sample with a half-life of 10 hours if its initial activity was 2000 Bq.

The activity $A(t)$ at time $t$ is given by:

$A(t) = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}$

where $A_0$ is the initial activity, $T$ is the half-life, and $t$ is the elapsed time.

For a sample with a half-life of 10 hours:

$A(t) = 2000 \left(\frac{1}{2}\right)^{\frac{t}{10}}$

If $t = 20$ hours:

$A(20) = 2000 \left(\frac{1}{2}\right)^{\frac{20}{10}} = 2000 \left(\frac{1}{2}\right)^2 = 2000 \times \frac{1}{4} = 500 \text{ Bq}$

5.3 Discuss the significance of the neutron-proton ratio in determining the stability of a nucleus.

The neutron-proton ratio (N/Z) is crucial in determining nuclear stability. A nucleus with a balanced ratio of neutrons to protons is generally more stable. For lighter elements, a ratio close to 1:1 is stable, while for heavier elements, a higher neutron-to-proton ratio is required to counteract the increasing electrostatic repulsion between protons. Deviations from this ratio can lead to radioactive decay as the nucleus seeks stability.

Conclusion

Understanding the key concepts of Chapter 13 "Nuclei" in Class 12 Physics is essential for mastering nuclear physics. By exploring the structure of the nucleus, types of radioactive decay, nuclear reactions, and theoretical models, students can gain a deeper insight into the fundamental principles governing atomic nuclei. Practice with important questions and calculations will further solidify their grasp of these concepts, preparing them for both academic examinations and real-world applications in the field of nuclear physics.