Nuclei Class 12 Physics Chapter 13 Notes

The nucleus is the central part of an atom where the positive charge and most of the mass are concentrated.
Experiments, particularly those involving the scattering of alpha particles, have revealed that the nucleus is much smaller in size compared to the entire atom.
The radius of a nucleus is approximately 10,000 times smaller than the radius of the atom, demonstrating that atoms are mostly empty space.
Despite its small size, the nucleus contains more than 99.9% of the mass of the atom.

Atomic Mass Unit (u):

The atomic mass unit (u) is a more convenient unit for measuring atomic masses compared to kilograms.
It is defined as 1/12th of the mass of a carbon-12 (12C) atom.
The mass of a carbon-12 atom is approximately 1.992647 × 10^(-26) kg, and this value is used to derive the atomic mass unit.
The mass of a proton (mp) is approximately 1.00727 u.

Isotopes:

Isotopes are atomic species of the same element that have the same chemical properties but differ in mass.
Practically every element consists of a mixture of several isotopes.
The relative abundance of isotopes varies from element to element.
For example, chlorine has two isotopes with masses of 34.98 u and 36.98 u, with relative abundances of 75.4% and 24.6%, respectively.
The average mass of a chlorine atom is calculated based on the weighted average of its isotopes, resulting in an atomic mass of 35.47 u.

Hydrogen Isotopes:

Hydrogen has three isotopes with masses of 1.0078 u, 2.0141 u, and 3.0160 u.
The lightest isotope, with a relative abundance of 99.985%, is called the proton.
The mass of a proton (mp) is approximately 1.00727 u.
The other two isotopes of hydrogen are deuterium and tritium, with tritium being unstable and not occurring naturally.

Composition of Nucleus:

The positive charge in the nucleus is carried by protons.
Protons carry one unit of fundamental charge (+e) and are stable.
Electrons are outside the nucleus, and their total charge is (–Ze), where Z is the atomic number.
To maintain overall neutrality, the nucleus contains an equal number of protons, which is exactly Z.

Atomic masses and composition of nucleus

Atomic Mass Unit (u):

The atomic mass unit (u) is a more convenient unit for measuring atomic masses compared to kilograms.
It is defined as 1/12th of the mass of a carbon-12 (12C) atom.
The mass of a carbon-12 atom is approximately 1.992647 × 10^(-26) kg, and this value is used to derive the atomic mass unit.
The mass of a proton (mp) is approximately 1.00727 u.

Isotopes:

Isotopes are atomic species of the same element that have the same chemical properties but differ in mass.
Practically every element consists of a mixture of several isotopes.
The relative abundance of isotopes varies from element to element.
For example, chlorine has two isotopes with masses of 34.98 u and 36.98 u, with relative abundances of 75.4% and 24.6%, respectively.
The average mass of a chlorine atom is calculated based on the weighted average of its isotopes, resulting in an atomic mass of 35.47 u.

Hydrogen Isotopes:

Hydrogen has three isotopes with masses of 1.0078 u, 2.0141 u, and 3.0160 u.
The lightest isotope, with a relative abundance of 99.985%, is called the proton.
The mass of a proton (mp) is approximately 1.00727 u.
The other two isotopes of hydrogen are deuterium and tritium, with tritium being unstable and not occurring naturally.

Composition of Nucleus:

The positive charge in the nucleus is carried by protons.
Protons carry one unit of fundamental charge (+e) and are stable.
Electrons are outside the nucleus, and their total charge is (–Ze), where Z is the atomic number.
To maintain overall neutrality, the nucleus contains an equal number of protons, which is exactly Z.

Discovery of Neutron

Deuterium and tritium nuclei, isotopes of hydrogen, have different masses despite having one proton each, suggesting the presence of neutral matter.
James Chadwick observed the emission of neutral radiation when beryllium nuclei were bombarded with alpha particles and hypothesized the existence of a new type of neutral particle, the neutron.
The mass of a neutron (mn) is approximately 1.00866 atomic mass units (u) or 1.6749 x 10^(-27) kg.

Properties of Neutron:

A free neutron is unstable and decays into a proton, an electron, and an antineutrino with a mean life of about 1000 seconds.
Inside the nucleus, neutrons are stable and contribute to the nucleus’s stability.

Nuclear Composition:

The composition of a nucleus is described using the following terms and symbols:
- Z: Atomic number, the number of protons.
- N: Neutron number, the number of neutrons.
- A: Mass number, the total number of protons and neutrons (A = Z + N).
- Nucleon: Term used for a proton or a neutron.

Nuclide Notation:

Nuclear species or nuclides are represented by the notation XAZ, where X is the chemical symbol of the species.
For example, the nucleus of gold is denoted as 19779Au, indicating that it contains 197 nucleons, including 79 protons and 118 neutrons.

Isotopes and Isobars:

Isotopes of an element have the same number of protons (Z) but differ in their number of neutrons (N).
Chemical properties of isotopes are identical because they have the same electronic structure and are placed in the same location in the periodic table.
Nuclides with the same mass number A are called isobars.
Nuclides with the same neutron number N but different atomic number Z are called isotones.

Size of the Nucleus

The classic experiment conducted by Geiger and Marsden involved the scattering of alpha particles (α-particles) from thin gold foils.
It was observed that the distance of closest approach of an α-particle with a kinetic energy of 5.5 MeV to a gold nucleus is about 4.0 × 10^(-14) meters.
Rutherford’s interpretation of this scattering was based on the Coulomb repulsion between the positive charges of the α-particle and the gold nucleus.
Scattering experiments with higher-energy α-particles, which lead to smaller distances of closest approach, can reveal deviations from Rutherford’s calculations and provide insights into nuclear sizes.
Scattering experiments using fast electrons have been conducted to measure the sizes of nuclei of various elements.

Nuclear Size Formula:

The size (radius) of a nucleus with a mass number A is given by R = R0 * A^(1/3), where R0 is approximately 1.2 × 10^(-15) meters (1.2 femtometers or 1.2 fm). (1 fm = 10^(-15) meters)
This formula implies that the volume of the nucleus is proportional to A, the mass number.
The density of the nucleus, which is constant for all nuclei, is approximately 2.3 × 10^(17) kg/m^3.
This nuclear density is significantly higher than the density of ordinary matter, such as water (1,000 kg/m^3), highlighting that most of an atom is empty space.

Mass-Energy Equivalence

Albert Einstein’s theory of special relativity introduced the concept that mass and energy are interconnected and can be converted into each other.
Before this theory, it was commonly assumed that mass and energy were conserved separately in physical reactions.
Einstein’s groundbreaking insight was that mass can be considered as another form of energy, and it is possible to convert mass into other forms of energy (e.g., kinetic energy) and vice versa.
The famous mass-energy equivalence relation, often written as E = mc^2, quantifies this relationship, where:
- E represents energy.
- m represents mass.
- c represents the speed of light in a vacuum, which is approximately equal to 3 × 10^8 meters per second (m/s).

Experimental Verification:

Experimental confirmation of Einstein’s mass-energy relation has been achieved in the study of various physical processes and reactions, particularly in the domain of nuclear physics.
In any physical reaction, the conservation of energy states that the total energy before the reaction should be equal to the total energy after the reaction.
This principle of energy conservation includes the recognition that mass itself carries an associated energy, as given by E = mc^2.

Significance:

The mass-energy equivalence relation has profound implications for understanding nuclear masses and the interactions of nuclei with one another.
It has provided crucial insights into the behavior of particles in nuclear reactions and the release of energy in processes like nuclear fusion and fission.

Nuclear binding energy

Nuclear Mass and Mass Defect:

The nucleus of an atom is composed of protons and neutrons. However, the mass of the nucleus (M) is found to be less than the sum of the masses of its individual protons and neutrons.
This mass difference between the nucleus and the sum of its constituents is referred to as the “mass defect,” denoted as ΔM. The mass defect arises because of Einstein’s mass-energy equivalence.

Einstein’s Mass-Energy Equivalence:

According to Einstein’s famous equation, E = mc^2, mass and energy are interchangeable. In the context of nuclei, the mass defect represents energy that is associated with the nucleus but is not present in the sum of the masses of its individual particles.

Binding Energy:

The energy equivalent of the mass defect, ΔM, is the energy required to break the nucleus into its individual protons and neutrons. This energy is known as the “binding energy” (Eb) of the nucleus.
The binding energy is the amount of energy that holds the nucleons (protons and neutrons) together within the nucleus.
The relationship between the mass defect and the binding energy is given by the equation Eb = ΔMc^2.

Binding Energy per Nucleon:

The binding energy per nucleon (Ebn) is defined as the ratio of the binding energy of a nucleus (Eb) to the number of nucleons (A) in that nucleus. It represents the average energy required to separate a nucleus into its individual nucleons.
Ebn = Eb / A

Observations from the Binding Energy per Nucleon vs. Mass Number Plot:

The text describes a plot of the binding energy per nucleon (Ebn) versus the mass number (A) for various nuclei.
Notable observations from the plot include:
1. Ebn is nearly constant for nuclei with intermediate mass numbers (around 30 < A < 170), with a maximum value of about 8.75 MeV at A = 56.
2. Ebn is lower for both lighter nuclei (A < 30) and heavier nuclei (A > 170).
3. The constant value of Ebn within a specific range is attributed to the short-range nature of the nuclear force.
4. When very heavy nuclei break into two lighter nuclei, nucleons become more tightly bound, releasing energy (fission).
5. Conversely, when light nuclei fuse to form a heavier nucleus, the final system becomes more tightly bound, releasing energy (fusion).

Significance:

The concept of nuclear binding energy is fundamental in understanding the stability and behavior of atomic nuclei.
It has significant implications for energy production, such as in nuclear fission and fusion processes, as well as in astrophysical contexts like the energy source of the sun.

Strength of the Nuclear Force:

The nuclear force is extraordinarily strong, much stronger than the electromagnetic (Coulomb) force between electric charges or the gravitational force between masses.
It needs to be significantly stronger than the Coulomb repulsion between positively charged protons within the nucleus to hold the nucleus together.

Short-Range Nature of the Nuclear Force:

The nuclear force operates only at extremely short distances on the order of a few femtometers (1 femtometer = 10^(-15) meters).Beyond a few femtometers, the nuclear force rapidly falls to zero. This property leads to the saturation of forces within medium to large-sized nuclei.
Saturation results in the constancy of the binding energy per nucleon, as nucleons inside the nucleus interact primarily with their closest neighbors.

Repulsive and Attractive Nature:

The potential energy between two nucleons is characterized by a minimum value at a specific distance (approximately 0.8 femtometers), denoted as r0.
At distances greater than r0, the nuclear force is attractive, effectively drawing nucleons together.
At distances less than r0, the nuclear force becomes repulsive, preventing nucleons from coming too close.

Independence of Electric Charge:

The strength of the nuclear force is approximately the same for neutron-neutron, proton-neutron, and proton-proton interactions.
Unlike the Coulomb force, the nuclear force is not influenced by the electric charge of the particles involved.

Lack of Simple Mathematical Form:

Unlike the well-defined mathematical expressions for forces like Coulomb’s law or Newton’s law of gravitation, the nuclear force lacks a simple mathematical form. It is described by complex interactions and cannot be represented by a straightforward formula.

Radioactivity is a fascinating and significant nuclear phenomenon, and it was discovered by A. H. Becquerel in 1896 under unexpected circumstances. He made the discovery while studying fluorescence and phosphorescence in compounds exposed to visible light. The discovery is summarized as follows:

Discovery of Radioactivity

A. H. Becquerel was investigating the behavior of compounds when exposed to visible light.
He chose uranium-potassium sulphate for his experiments.
After illuminating samples of this compound with visible light, he observed a remarkable occurrence.
Becquerel placed these illuminated samples inside a light-tight package made of black paper.
To separate this package from a photographic plate, he added a piece of silver between them.
The intention was to see if any effect from the compound could pass through both the black paper and silver.
After several hours of exposure, when the photographic plate was developed, it displayed a darkening or blackening.
The darkening on the photographic plate indicated that something had been emitted by the uranium-potassium sulphate compound, and this emitted radiation had the capability to penetrate both the black paper and the silver.

Subsequent experiments confirmed that this phenomenon was related to nuclear processes occurring within the substance, marking the discovery of radioactivity. Radioactive decay, the core principle behind radioactivity, involves unstable nuclei undergoing various types of decay processes. There are three main types of radioactive decay:

Types of Radioactive Decay:

Alpha Decay (α-decay): In this type of decay, a helium nucleus (4 2He) is emitted from the radioactive nucleus.
Beta Decay (β-decay): Beta decay involves the emission of electrons or positrons (particles with the same mass as electrons but opposite charge).
Gamma Decay (γ-decay): Gamma decay is characterized by the emission of high-energy photons, typically with energies in the hundreds of kiloelectronvolts (keV) or more.

These types of radioactive decay play a crucial role in understanding nuclear physics and are fundamental to the behavior of unstable nuclei. Each type of decay has unique characteristics, and they have diverse applications in various fields of science and technology.

Binding Energy and Nuclear Energy

The binding energy per nucleon (Ebn) curve in Figure 13.1 reveals a significant feature. In the middle region, roughly between A = 30 and A = 170, the binding energy per nucleon is nearly constant at about 8.0 MeV (megaelectronvolts).
Nuclei with lower total binding energy can release energy when they transform into nuclei with greater binding energy. This energy release results from the conversion of mass into energy.
Two fundamental nuclear processes harness this energy release: fission and fusion.

Fission:

Fission is the process where a heavy nucleus, typically with a large number of nucleons, splits into two or more intermediate mass fragments. This process results in a net energy release.
For example, when a heavy nucleus like uranium undergoes fission, it transforms into smaller nuclei, releasing a significant amount of energy. This process is used in nuclear reactors and atomic bombs.
Fission reactions involve a substantial energy release, which is on the order of MeV (megaelectronvolts).

Fusion:

Fusion is the process in which light nuclei combine to form a heavier nucleus. This process is the basis of the energy generation in stars like our Sun.
Fusion reactions are responsible for the release of energy and are commonly considered in the context of nuclear fusion as a potential future energy source.
Fusion reactions also yield a considerable energy release, typically in the range of MeV.

Comparing Nuclear and Chemical Reactions:

One significant distinction between nuclear and chemical reactions is the energy scale involved. Chemical reactions, such as burning coal or petroleum, release energy in the range of electron volts (eV).
In contrast, nuclear reactions, whether through fission or fusion, release energy on the order of MeV. This means nuclear sources can generate a million times more energy per unit of matter compared to chemical sources.
For example, when 1 kilogram of uranium undergoes fission, it produces a massive 10^14 joules (J) of energy. In contrast, burning 1 kilogram of coal yields approximately 10^7 J. The energy difference is substantial.

Fission Reaction

An example of a fission reaction involves bombarding a uranium isotope, such as 235U, with a neutron. This interaction results in the uranium nucleus breaking apart into two intermediate mass nuclear fragments. The reaction can be represented as follows: 235/92(U)+1/0(n)→92/36(Kr)+144/56(Ba)+3*1/0(n)
This process is accompanied by the release of several neutrons, which can further induce fission reactions in nearby uranium nuclei, resulting in a chain reaction.

Variability in Fission Products:

The fission of a uranium nucleus can produce different pairs of intermediate mass fragments. For example, it may yield 99/41(Nb) + 133/51(Sb), or 94/38(Sr) + 140/54(Xe), among others.
These intermediate mass fragments are typically radioactive and may emit beta particles (β) to achieve stable end products.

Energy Release in Fission:

The energy released in a fission reaction is substantial. The Q-value (energy release) for fission reactions of heavy nuclei like uranium is approximately 200 MeV (megaelectronvolts) per fissioning nucleus.
To estimate this energy release, consider a nucleus with mass number A = 240 breaking into two fragments, each with A = 120.
The binding energy per nucleon (Ebn) for the A = 240 nucleus is around 7.6 MeV, while the Ebn for the two A = 120 fragment nuclei is approximately 8.5 MeV.
The gain in binding energy per nucleon is about 0.9 MeV. Thus, the total gain in binding energy for the entire fission process is approximately 240 nucleons × 0.9 MeV/nucleon, which is about 216 MeV.
The energy released during fission initially appears as the kinetic energy of the fragments and neutrons. Eventually, it is converted into heat, which can be harnessed for various purposes.

Applications of Fission:

Nuclear reactors use controlled fission reactions to generate electricity. In these reactors, the energy released from nuclear fission is employed to produce steam, which drives turbines to generate electrical power.
In contrast, uncontrolled nuclear fission reactions are the source of the enormous energy released in atomic bombs (nuclear weapons).

Nuclear Fusion Reactions

Nuclear fusion reactions release energy when two light nuclei merge to form a heavier nucleus with higher binding energy. Some examples of energy-releasing nuclear fusion reactions include:
1. Proton-proton chain reaction (pp-chain) in stars, which converts hydrogen into helium:
2. 11H+11H→12H+e++νe+0.42MeV(Reaction (i))
3. e++e−→γ+γ+1.02MeV(Reaction (ii))
4. 12H+11H→23He+γ+5.49MeV(Reaction (iii))
5. 23He+23He→24He+11H+11H+12.86MeV(Reaction (iv))
These reactions collectively convert four hydrogen nuclei into one helium nucleus with the release of a total energy of 26.7 MeV.

Thermonuclear Fusion:

Fusion reactions occur in stars when the temperature is sufficiently high for particles to overcome the Coulomb repulsion between positively charged nuclei. This process, where temperature is the driving factor, is termed thermonuclear fusion.
In stars like the sun, the core temperature (about 15 million degrees Celsius or 27 million degrees Fahrenheit) is sufficient to initiate hydrogen-to-helium fusion.

Star’s Hydrogen-Burning Process:

In the sun and stars like it, the primary source of energy is the hydrogen-to-helium fusion process. The proton-proton (pp) cycle drives the conversion of hydrogen into helium.
This hydrogen-burning process is a multi-step sequence of reactions, as described above. It is this process that generates the sun’s energy.

Stellar Evolution:

As a star’s core becomes primarily helium, it starts to cool and contract due to gravity. As the core temperature increases to around 108108 K, helium-to-carbon fusion reactions begin.
These fusion processes are responsible for the production of higher-mass elements within stars. However, elements heavier than those near the peak of the binding energy curve cannot be produced through this mechanism.

The Sun’s Future:

The sun is about 5 billion years old and is expected to burn hydrogen for another 5 billion years.
After the hydrogen-burning phase, the sun will transition to a red giant as it begins to cool and collapse due to gravitational forces.

Controlled Thermonuclear Fusion

Controlled thermonuclear fusion is an ambitious and promising approach to harnessing nuclear fusion as a clean and virtually limitless source of energy. Here are the key points regarding controlled thermonuclear fusion:

Objective of Controlled Thermonuclear Fusion:

The primary objective of controlled thermonuclear fusion is to replicate the natural fusion processes that occur in stars, such as our sun, in a controlled environment on Earth.
The goal is to achieve a sustained and controlled fusion reaction that can produce a steady and abundant source of energy for various applications, including electricity generation.

Operating Conditions:

In a controlled fusion reactor, the nuclear fuel is typically composed of isotopes of light elements, such as deuterium (a heavy hydrogen isotope) and tritium (another hydrogen isotope). These isotopes are heated to extremely high temperatures, typically around 108108 Kelvin, which creates a state of matter called a plasma.
In this high-temperature plasma, the atoms are stripped of their electrons, leaving a mixture of positively charged ions and free electrons. This plasma state is essential for initiating nuclear fusion reactions.

Challenges:

One of the most significant challenges in controlled thermonuclear fusion is confining and stabilizing the extremely hot plasma. The plasma’s temperature is so high that no material container can withstand it.
Researchers and engineers have developed various techniques to contain the plasma and prevent it from coming into contact with the walls of the fusion chamber. These methods include magnetic confinement using devices like tokamaks and stellarators, as well as inertial confinement through laser-driven implosions.

Global Efforts:

Controlled thermonuclear fusion research is a global endeavor, with numerous countries investing in the development of fusion technologies. Projects such as ITER (International Thermonuclear Experimental Reactor) in France are international collaborations designed to demonstrate the feasibility of sustained nuclear fusion and its potential for practical energy production.
India is also actively involved in fusion research and development through initiatives like the “Steady State Tokamak (SST-1)” and “ITER-India.”

Prospects:

The successful development of controlled thermonuclear fusion holds the promise of providing a nearly limitless, safe, and environmentally clean source of energy.
Fusion reactors could potentially offer a solution to the world’s growing energy needs while reducing greenhouse gas emissions and dependence on fossil fuels.