Nuclei – CBSE NCERT Study Resources

The binding energy per nucleon curve plots the average energy required to remove a nucleon from a nucleus. It peaks around iron-56, indicating maximum stability.

• For lighter nuclei (e.g., hydrogen), fusion increases binding energy, releasing energy (e.g., Sun's energy).
• For heavier nuclei (e.g., uranium-235), fission increases binding energy, releasing energy (e.g., nuclear reactors).

Our textbook shows iron-56 as the most stable, explaining why fusion occurs in stars and fission in reactors.

This principle guides sustainable energy research, like ITER for fusion power.

Nuclear force is the strong, short-range (<1 fm) attraction between nucleons, overcoming electrostatic repulsion between protons.

• It is 100× stronger than electrostatic force but acts only within nuclei (e.g., helium-4 stability).
• Gravity is negligible at nuclear scales (e.g., neutron stars show dominance of nuclear force).

We studied that without nuclear force, nuclei would disintegrate due to proton repulsion.

Understanding it aids in quark-gluon plasma research at CERN.

The decay law states N = N₀e^-λt, where λ is the decay constant. Half-life (T_½) is ln(2)/λ.

• Carbon-14 (T_½ = 5730 years) decays predictably, enabling age estimation (e.g., fossils).
• Example: A sample with 25% C-14 implies two half-lives (≈11,460 years).

Our textbook shows logarithmic decay fits observed data precisely.

Accurate dating refines archaeological and geological timelines.

Einstein's E=mc² states mass defect (Δm) converts to energy (E=Δm×931.5 MeV/u).

• D + T → He + n: Δm = (2.0141+3.0160) − (4.0026+1.0087) = 0.0188u.
• Energy = 0.0188 × 931.5 = 17.5 MeV (e.g., ITER reactor fuel).

We studied this in the context of the Sun’s energy output.

Fusion energy could replace fossil fuels with minimal waste.

The shell model proposes nucleons occupy discrete energy levels, with magic numbers (2, 8, 20, etc.) indicating filled shells.

• Double-magic nuclei (e.g., helium-4, oxygen-16) show high stability and abundance.
• Lead-208 (Z=82, magic) is the heaviest stable nucleus.

Our textbook correlates magic numbers with low decay rates and high binding energy.

This model guides synthesis of superheavy elements like oganesson (Z=118).

The decay law states dN/dt=−λN, where λ is the decay constant. Integrated form: N=N₀e^−λt.

• Half-life (t_1/2) is when N=N₀/2, giving t_1/2=ln2/λ.
• Experimentally, Geiger-Müller counters track decay rates over time (e.g., C-14 dating).

We studied that λ is unique for each isotope (e.g., U-238: 4.5×10⁹ years). This predictability enables applications like nuclear medicine.

Accurate half-life measurements are vital for radiation therapy and archaeological dating.

Fusion combines light nuclei (e.g., ²H+³H→⁴He+n), while fission splits heavy ones (e.g., ²³⁵U+n→fragments).

Fusion produces minimal waste (helium) versus fission's radioactive fragments (e.g., Sr-90). However, fusion remains experimentally challenging (ITER project).

Fusion could offer clean energy if confinement issues are solved.

The model treats the nucleus as an incompressible liquid drop. The mass formula includes:

• Volume energy (A)
• Surface energy (−A^2/3)
• Coulomb repulsion (−Z²/A^1/3)

Our textbook shows binding energy B=a_vA−a_sA^2/3−a_cZ²/A^1/3+pairing terms. This matches observed nuclear masses within 1%.

While successful for medium nuclei, it fails to explain magic numbers, addressed by the shell model.

The model remains foundational for predicting nuclear properties in astrophysics.

Pauli proposed neutrinos (ν) to conserve energy in β-decay (n→p+e⁻+ν̄_e). They carry missing energy/momentum.

• Reines-Cowan experiment (1956) detected ν̄_e via ν̄_e+p→n+e⁺.
• Modern detectors like Super-Kamiokande observe solar neutrinos.

We studied that neutrinos have tiny mass (<0.1 eV/c²) and oscillate between flavors, proving beyond Standard Model physics.

They're key to understanding supernovae and dark matter candidates.

The binding energy per nucleon is the energy required to disassemble a nucleus into its constituent protons and neutrons. It is a measure of nuclear stability. Our textbook shows that the curve peaks around iron-56 (A=56), indicating maximum stability.

• For light nuclei (A < 20), the curve rises sharply due to strong nuclear forces overcoming Coulomb repulsion.
• For intermediate nuclei (20 < A < 60), the curve is nearly flat, showing optimal stability.
• For heavy nuclei (A > 60), the curve declines due to increasing Coulomb repulsion.

This explains why fusion occurs in stars (light nuclei) and fission occurs in heavy nuclei like uranium-235. The curve validates the semi-empirical mass formula.

Understanding this curve helps in designing nuclear reactors and predicting energy release in fusion/fission processes.

Nuclear fusion is the process where lighter nuclei combine to form heavier nuclei, releasing energy. In stars like the Sun, the proton-proton chain is the dominant fusion mechanism.

• Step 1: Two protons fuse to form deuterium, a positron, and a neutrino.
• Step 2: Deuterium fuses with another proton to form helium-3.
• Step 3: Two helium-3 nuclei combine to form helium-4, releasing two protons.

This process converts 0.7% of mass into energy (E=mc²). Our textbook shows it produces 26.7 MeV per helium-4 nucleus, sustaining solar energy.

Replicating controlled fusion on Earth could provide clean, limitless energy, as seen in ITER experiments.

Fission splits heavy nuclei (e.g., uranium-235), while fusion combines light nuclei (e.g., hydrogen). Both release energy via mass defect (E=mc²).

Fission is commercially viable but produces radioactive waste. Fusion is cleaner but technologically challenging.

Fusion could revolutionize energy if confinement and sustainability issues are resolved.

The decay law states dN/dt = -λN, where λ is the decay constant. Integrating gives N = N₀e^(-λt), showing exponential decay.

• Half-life (T½): Time for half the nuclei to decay. At t = T½, N = N₀/2. Substituting, we get T½ = ln(2)/λ ≈ 0.693/λ.
• Mean life (τ): Average lifetime of a nucleus. Calculated as τ = 1/λ.

Thus, T½ = τ·ln(2). For carbon-14 (T½=5730 years), τ≈8267 years. Our textbook confirms this via decay simulations.

These concepts are vital in radiometric dating (e.g., carbon dating) and nuclear medicine (e.g., half-life of isotopes).

The binding energy per nucleon is the energy required to disassemble a nucleus into its constituent protons and neutrons. It peaks around iron-56, indicating maximum stability.

• For heavy nuclei (e.g., uranium-235), fission releases energy as products have higher binding energy per nucleon.
• For light nuclei (e.g., hydrogen), fusion increases binding energy per nucleon, releasing energy.

Our textbook shows the curve's peak explains why iron-56 is the most stable. Deviations indicate energy release potential.

This principle underpins nuclear power and stellar energy production, highlighting its universal relevance.

The decay law states N = N₀e^-λt, where λ is the decay constant. Half-life (T_1/2) is derived as ln(2)/λ.

• Geiger-Müller counters measure activity (e.g., for cobalt-60), showing exponential decay.
• Plotting ln(N) vs time yields a straight line, validating the law.

We studied that half-life is independent of initial quantity, confirmed by consistent T_1/2 in experiments.

This law is crucial for radiometric dating and medical applications like radiotherapy.

Fission splits heavy nuclei (e.g., uranium-235), while fusion combines light nuclei (e.g., hydrogen isotopes).

• Fission: Requires neutron bombardment, releases 200 MeV/event (e.g., nuclear reactors).
• Fusion: Needs high temperature/pressure (e.g., sun, ITER project), yields ~17 MeV per reaction.

Our textbook shows fusion is cleaner but technologically challenging compared to fission.

Fusion could offer limitless energy if confinement challenges are overcome.

Mass defect is the difference between a nucleus's mass and its nucleons' total mass. Einstein's E=mc² converts this to energy.

• In the sun, 4 hydrogen nuclei (mass ~4.032 u) fuse to helium (4.0026 u), releasing 26.73 MeV.
• This 0.7% mass defect powers stars via proton-proton chain.

We studied that even tiny mass defects yield enormous energy, validating nuclear astrophysics.

Harnessing this process on Earth could revolutionize energy production.

The nuclear binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons. It is a measure of the stability of the nucleus, as a higher binding energy indicates a more stable nucleus. The binding energy arises due to the strong nuclear force, which overcomes the electrostatic repulsion between protons.

The binding energy (B.E.) can be calculated using the mass defect (Δm) and Einstein's mass-energy equivalence formula: B.E. = Δm × c², where c is the speed of light. The mass defect is the difference between the mass of the separated nucleons and the actual mass of the nucleus.

To derive the binding energy per nucleon, we divide the total binding energy by the mass number (A): B.E. per nucleon = B.E. / A. This value indicates the average energy needed to remove a nucleon from the nucleus.

The binding energy per nucleon varies with the mass number as follows:

• For light nuclei (A < 30), the binding energy per nucleon increases rapidly due to the increasing dominance of the strong nuclear force.
• For medium nuclei (30 < A < 90), it reaches a peak (around 8.8 MeV for iron-56), indicating maximum stability.
• For heavy nuclei (A > 90), it gradually decreases due to the increasing Coulomb repulsion between protons.

This variation explains why nuclear fusion occurs in light nuclei (to increase binding energy) and fission occurs in heavy nuclei (to achieve more stable configurations).

The radioactive decay law states that the rate of decay of radioactive nuclei is proportional to the number of undecayed nuclei present at any time. Mathematically, it is expressed as: dN/dt = -λN, where N is the number of undecayed nuclei, t is time, and λ is the decay constant.

To derive the expression for half-life (T_1/2), we start by solving the decay law equation. Integrating both sides gives: N = N₀e^-λt, where N₀ is the initial number of nuclei.

Half-life is defined as the time taken for half of the radioactive nuclei to decay. At t = T_1/2, N = N₀/2. Substituting into the decay equation:

Simplifying, we get:

Taking the natural logarithm on both sides:

Which simplifies to:

This shows that the half-life is inversely proportional to the decay constant. A smaller decay constant (λ) means a longer half-life, indicating a slower decay process.

This relationship is crucial in applications like radiometric dating and nuclear medicine, where knowing the half-life helps predict the decay behavior of radioactive substances.

Nuclear fission is the process where a heavy nucleus, such as uranium-235 or plutonium-239, splits into smaller nuclei when bombarded with a neutron, releasing a large amount of energy. For example, when a neutron strikes a uranium-235 nucleus, it splits into barium-141 and krypton-92, along with the emission of three neutrons and energy.

The chain reaction occurs when the neutrons released during fission strike other uranium-235 nuclei, causing them to split and release more neutrons. This self-sustaining process is crucial for controlled energy release in nuclear reactors.

In nuclear power plants, the chain reaction is carefully controlled using control rods (like cadmium or boron) to absorb excess neutrons, ensuring a steady energy output. This process is significant because it provides a large amount of energy from a small amount of fuel, reducing reliance on fossil fuels and minimizing greenhouse gas emissions.

The binding energy per nucleon curve is a plot of the binding energy per nucleon against the mass number (A) of nuclei. It peaks around iron-56, indicating that nuclei with mass numbers near iron are the most stable.

Implications:

• For nuclear fission, heavy nuclei (like uranium) split into smaller fragments closer to the peak of the curve, releasing energy because the binding energy per nucleon increases.
• For nuclear fusion, light nuclei (like hydrogen) combine to form heavier nuclei (like helium), moving toward the peak of the curve and releasing energy.

This curve explains why both fusion and fission release energy. Fusion occurs in stars, including the Sun, where hydrogen nuclei fuse to form helium. Fission is used in nuclear reactors, where heavy nuclei split to produce energy. The curve highlights the stability of intermediate-mass nuclei and the energy potential of nuclear reactions.

Nuclear fission is the process where a heavy nucleus, such as uranium-235 or plutonium-239, splits into smaller nuclei when bombarded with a neutron, releasing a large amount of energy. For example, when a neutron strikes a uranium-235 nucleus, it splits into barium-141 and krypton-92, along with the emission of three neutrons and energy.

The chain reaction occurs when the neutrons released during fission strike other uranium-235 nuclei, causing them to split and release more neutrons. This self-sustaining process multiplies the energy output exponentially.

The significance of chain reactions in energy production lies in their ability to release massive amounts of energy from a small amount of fuel. Controlled chain reactions in nuclear reactors are used to generate electricity, while uncontrolled reactions lead to nuclear explosions.

Additional Information: The energy released in fission is due to the conversion of a small amount of mass into energy, as described by Einstein's mass-energy equivalence (E=mc²).

The binding energy per nucleon curve plots the average energy required to remove a nucleon from a nucleus against the mass number. It peaks around iron-56, indicating that nuclei near this region are the most stable.

Implications on nuclear stability:

• Nuclei with low mass numbers (e.g., hydrogen, helium) have lower binding energy per nucleon, making them less stable.
• Nuclei with mass numbers near iron-56 have higher binding energy, making them highly stable.
• Heavy nuclei (e.g., uranium) have slightly lower binding energy, making them prone to fission.

In nuclear fusion, lighter nuclei combine to form a heavier nucleus closer to the peak of the curve, releasing energy due to the increase in binding energy per nucleon. For example, fusion in stars converts hydrogen into helium, releasing vast energy.

In nuclear fission, heavy nuclei split into smaller fragments closer to the peak, releasing energy because the products have higher binding energy per nucleon than the original nucleus.

Additional Information: The curve explains why fusion is dominant in stars (light nuclei) and fission is used on Earth (heavy nuclei) for energy production.

Nuclear fission is the process in which a heavy nucleus, such as uranium-235 or plutonium-239, splits into two or more smaller nuclei, along with the release of a large amount of energy and several neutrons. This occurs when the nucleus absorbs a slow-moving neutron, becomes unstable, and splits.

For example, when a uranium-235 nucleus absorbs a neutron, it forms uranium-236, which is highly unstable. It then splits into two smaller nuclei like barium-141 and krypton-92, along with the release of three neutrons and a tremendous amount of energy (approximately 200 MeV per fission).

The released neutrons can further cause fission in other uranium-235 nuclei, leading to a chain reaction. For a sustained chain reaction, the critical mass of the fissile material must be achieved, ensuring that enough neutrons are available to continue the process.

Applications in energy production include:

• Nuclear power plants: Controlled chain reactions are used to generate heat, which produces steam to drive turbines and generate electricity.
• Nuclear weapons: Uncontrolled chain reactions result in massive energy release, as seen in atomic bombs.

Understanding nuclear fission and chain reactions is crucial for harnessing nuclear energy safely and efficiently, contributing to clean energy solutions.

Nuclear fusion is the process by which lighter nuclei combine to form heavier nuclei, releasing a tremendous amount of energy. In stars like the Sun, fusion occurs primarily through two processes: the proton-proton (p-p) chain and the carbon-nitrogen-oxygen (CNO) cycle.

Proton-Proton Chain:
1. Two protons (hydrogen nuclei) fuse to form a deuterium nucleus, releasing a positron and a neutrino.
2. The deuterium nucleus fuses with another proton to form helium-3, releasing a gamma-ray photon.
3. Two helium-3 nuclei combine to form helium-4, releasing two protons.

CNO Cycle:
1. A proton fuses with a carbon-12 nucleus to form nitrogen-13, releasing a gamma-ray photon.
2. Nitrogen-13 decays into carbon-13, emitting a positron and a neutrino.
3. Carbon-13 fuses with another proton to form nitrogen-14.
4. Nitrogen-14 captures another proton to form oxygen-15, which decays into nitrogen-15.
5. Nitrogen-15 fuses with a proton to produce carbon-12 and helium-4, completing the cycle.

Energy Release:
The mass of the resulting nucleus is slightly less than the sum of the original nuclei. This mass defect is converted into energy as per Einstein's equation E=mc², where E is energy, m is mass, and c is the speed of light. This energy sustains the star's luminosity and heat.

Nuclear fission is the process where a heavy nucleus, such as uranium-235 (U-235) or plutonium-239 (Pu-239), splits into smaller nuclei when bombarded with a neutron, releasing a large amount of energy. For example, when U-235 absorbs a slow neutron, it becomes unstable and splits into barium-141 and krypton-92, along with 2-3 neutrons and energy.

The released neutrons can further cause fission in other U-235 nuclei, leading to a chain reaction. This self-sustaining process multiplies the energy output exponentially if left uncontrolled.

In a nuclear reactor, the chain reaction is controlled using:

• Control rods (made of cadmium or boron) to absorb excess neutrons.
• Moderators (like heavy water or graphite) to slow down neutrons, ensuring efficient fission.
• Coolants to regulate temperature and transfer heat for energy production.

This controlled fission ensures steady energy release without explosions, making it useful for power generation.

Nuclear fusion is the process by which lighter nuclei combine to form heavier nuclei, releasing a tremendous amount of energy. In stars like the Sun, the primary mechanism is the proton-proton (p-p) chain, while in more massive stars, the CNO cycle dominates.

Proton-Proton Chain:

• Two protons fuse to form a deuterium nucleus, releasing a positron and a neutrino.
• The deuterium nucleus fuses with another proton to form helium-3.
• Two helium-3 nuclei combine to form helium-4, releasing two protons.

CNO Cycle:

• Carbon-12 acts as a catalyst, capturing protons to form nitrogen and oxygen isotopes.
• Eventually, helium-4 is produced, and carbon-12 is regenerated.

These processes convert mass into energy as per Einstein's mass-energy equivalence (E=mc²), sustaining the star's luminosity and heat. The energy released counteracts gravitational collapse, maintaining stellar equilibrium.

A nuclear reactor harnesses energy from controlled nuclear fission of uranium-235 or plutonium-239. The key components and their roles are:

1. Fuel Rods: Contain fissile material (e.g., U-235) where fission occurs, releasing neutrons and energy.

2. Moderator: (e.g., graphite or heavy water) slows down fast neutrons to thermal energies, increasing the likelihood of further fission.

3. Control Rods: (e.g., cadmium or boron) absorb excess neutrons to regulate the reaction rate and prevent runaway chain reactions.

4. Coolant: (e.g., water or liquid sodium) transfers heat from the reactor core to a steam generator, producing electricity.

Diagram (Conceptual):

• Core: Contains fuel rods immersed in moderator.
• Control rods inserted/withdrawn to adjust reactivity.
• Coolant circulates through the core to extract heat.

The reactor operates in a critical state, where the neutron production rate equals the loss rate, ensuring stable energy output. Safety systems prevent meltdowns by emergency shutdown mechanisms.