Class 12 Physics Notes Chapter 13 (Nuclei) – Examplar Problems (English) Book

Detailed Notes with MCQs of Chapter 13, 'Nuclei'. This is a crucial chapter, not just for your board exams but also frequently tested in various government competitive exams. We need a solid understanding of the concepts and formulas here. Pay close attention to the details.

Chapter 13: Nuclei - Detailed Notes for Exam Preparation

1. Introduction & Composition of Nucleus

• Nucleus: The central core of an atom, containing protons and neutrons. Discovered by Rutherford through his alpha-scattering experiment.
• Nucleons: Protons (positively charged) and neutrons (neutral) are collectively called nucleons.
• Atomic Number (Z): Number of protons in the nucleus. It determines the chemical element.
• Mass Number (A): Total number of nucleons (protons + neutrons) in the nucleus. A = Z + N.
• Neutron Number (N): Number of neutrons in the nucleus. N = A - Z.
• Nuclide Notation: An atom's nucleus is represented as ᴬ<0xE1><0xB5><0xA_Z>X, where X is the chemical symbol of the element. Example: ¹²₆C (Carbon-12).

2. Atomic Masses and Nuclear Species

• Atomic Mass Unit (amu or u): Defined as 1/12th the mass of an unperturbed Carbon-12 atom (¹²₆C) in its ground state.
  • 1 u = 1.660539 × 10⁻²⁷ kg
• Isotopes: Nuclides having the same atomic number (Z) but different mass numbers (A). They belong to the same element and have similar chemical properties. Example: ¹₁H (Protium), ²₁H (Deuterium), ³₁H (Tritium).
• Isobars: Nuclides having the same mass number (A) but different atomic numbers (Z). They belong to different elements. Example: ⁴⁰₁₈Ar and ⁴⁰₂₀Ca.
• Isotones: Nuclides having the same number of neutrons (N = A - Z) but different atomic numbers (Z) and mass numbers (A). Example: ³₁H (N=2) and ⁴₂He (N=2).

3. Size and Density of the Nucleus

• Nuclear Radius (R): Experimentally found to be proportional to the cube root of the mass number (A).
  • R = R₀ A¹ᐟ³
  • Where R₀ (empirical constant) ≈ 1.2 × 10⁻¹⁵ m = 1.2 fm (femtometre or fermi).
• Nuclear Volume (V): Assuming a spherical nucleus, V = (4/3)πR³ = (4/3)π(R₀ A¹ᐟ³ )³ = (4/3)πR₀³ A.
  • Volume is directly proportional to the mass number A.
• Nuclear Density (ρ): Mass per unit volume.
  • Mass of nucleus ≈ A × (mass of one nucleon) ≈ A × m (where m ≈ mass of proton ≈ mass of neutron ≈ 1.67 × 10⁻²⁷ kg).
  • ρ = Mass / Volume ≈ (A × m) / [(4/3)πR₀³ A] = 3m / (4πR₀³)
  • Notice that 'A' cancels out. This implies that nuclear density is nearly constant for all nuclei and is independent of the mass number A.
  • Its value is extremely high, approximately 2.3 × 10¹⁷ kg/m³.

4. Mass-Energy Equivalence & Nuclear Binding Energy

• Einstein's Mass-Energy Relation: E = mc² (where c is the speed of light in vacuum, c ≈ 3 × 10⁸ m/s). Mass and energy are inter-convertible.
• Energy Equivalent of 1 amu (u):
  • E = (1 u) × c² = (1.660539 × 10⁻²⁷ kg) × (2.9979 × 10⁸ m/s)²
  • E ≈ 931.5 MeV (Mega electron Volts)
  • So, 1 u ≈ 931.5 MeV/c² (This conversion is frequently used).
• Mass Defect (Δm): The difference between the sum of the masses of constituent nucleons (in their free state) and the actual mass of the nucleus.
  • Mass of constituent nucleons = [Z × m<0xE1><0xB5><0xA_p> + (A - Z) × m<0xE1><0xB5><0x82>] (where m<0xE1><0xB5><0xA_p> is mass of proton, m<0xE1><0xB5><0x82> is mass of neutron).
  • Actual mass of nucleus = M<0xE1><0xB5><0x8A><0xE1><0xB5><0x98><0xE1><0xB5><0x84>
  • Δm = [Z m<0xE1><0xB5><0xA_p> + (A - Z) m<0xE1><0xB5><0x82>] - M<0xE1><0xB5><0x8A><0xE1><0xB5><0x98><0xE1><0xB5><0x84>
  • Mass defect is always positive (Δm > 0) because the mass of the nucleus is always less than the sum of the masses of its constituents.
• Nuclear Binding Energy (B.E. or E<0xE1><0xB5><0x87>): The energy equivalent of the mass defect. It represents the energy required to completely separate all the nucleons from the nucleus, or conversely, the energy released when nucleons combine to form the nucleus.
  • B.E. = Δm c²
  • B.E. (in MeV) = Δm (in u) × 931.5 MeV/u
• Binding Energy per Nucleon (B.E./A or E<0xE1><0xB5><0x87><0xE1><0xB5><0x82>): The average energy required to remove one nucleon from the nucleus. B.E./A = (Total Binding Energy) / (Mass Number A).
  • It is a measure of the stability of the nucleus. Higher B.E./A means a more stable nucleus.
• Binding Energy Curve: A plot of B.E./A versus Mass Number (A).
  • Features:
    • Low B.E./A for light nuclei (A < 20).
    • Sharp peaks for nuclei like ⁴He, ¹²C, ¹⁶O, indicating higher stability.
    • Broad maximum around A = 56 (Iron, Fe), with B.E./A ≈ 8.75 MeV. These are the most stable nuclei.
    • Gradual decrease for heavy nuclei (A > 170).
  • Significance:
    • Explains energy release in nuclear fission (heavy nucleus splits into lighter ones with higher B.E./A).
    • Explains energy release in nuclear fusion (light nuclei fuse into a heavier one with higher B.E./A).

5. Nuclear Force

• The strong force that binds protons and neutrons together within the nucleus, overcoming the electrostatic repulsion between protons.
• Properties:
  • Strongest Force: Much stronger than electromagnetic and gravitational forces within the nuclear range.
  • Short Range: Acts only over very short distances (≈ few fm). Becomes negligible beyond this range. Has a repulsive core at very short distances (< 0.8 fm).
  • Charge Independent: Acts equally between proton-proton, neutron-neutron, and proton-neutron pairs.
  • Saturation Property: A nucleon interacts only with its immediate neighbours, not all nucleons in the nucleus. This leads to B.E./A being roughly constant for most nuclei.
  • Spin Dependent: Depends on the relative orientation of the spins of the interacting nucleons.
  • Non-central Force: Has a component that depends on the orientation of nucleon spins relative to the line joining them (tensor component).

6. Radioactivity

• The spontaneous disintegration of unstable nuclei by emitting particles (alpha, beta) or electromagnetic radiation (gamma rays). Discovered by Henri Becquerel.
• Law of Radioactive Decay: The rate of disintegration (dN/dt) is directly proportional to the number of undecayed nuclei (N) present at that instant.
  • dN/dt = -λN (The negative sign indicates that N decreases with time).
  • λ is the decay constant or disintegration constant. It is characteristic of the radioactive substance. Unit: s⁻¹, min⁻¹, year⁻¹, etc.
• Exponential Decay Equation: Integrating the decay law, we get:
  • N(t) = N₀ e^(-λt)
  • Where N₀ is the initial number of nuclei at t=0, and N(t) is the number of undecayed nuclei at time t.
• Activity (R): The rate of decay, i.e., the number of disintegrations per unit time.
  • R = |dN/dt| = λN
  • R(t) = λN(t) = λ(N₀ e^(-λt)) = (λN₀) e^(-λt) = R₀ e^(-λt)
  • Where R₀ = λN₀ is the initial activity.
  • Units of Activity:
    • Becquerel (Bq): SI unit. 1 Bq = 1 disintegration per second.
    • Curie (Ci): Older unit. 1 Ci = 3.7 × 10¹⁰ Bq (activity of 1g of Radium-226).
• Half-Life (T½ or t₁<0xE2><0x82><0x8F>₂): The time taken for the number of radioactive nuclei (or activity) to reduce to half its initial value.
  • When t = T½, N(t) = N₀/2.
  • N₀/2 = N₀ e^(-λT½) => 1/2 = e^(-λT½) => 2 = e^(λT½)
  • Taking natural logarithm: ln(2) = λT½
  • T½ = ln(2) / λ = 0.693 / λ
• Mean Life or Average Life (τ): The average lifetime of all the nuclei in a radioactive sample.
  • τ = 1 / λ
  • Relation between T½ and τ: T½ = ln(2) τ ≈ 0.693 τ (So, T½ < τ).
• Number of nuclei remaining after 'n' half-lives: N = N₀ (1/2)ⁿ, where n = t / T½.

7. Types of Radioactive Decay

• Alpha (α) Decay: Emission of an alpha particle (Helium nucleus, ⁴₂He). Occurs mainly in heavy nuclei (A > 210).
  • ᴬ<0xE1><0xB5><0xA_Z>X → ᴬ⁻⁴<0xE1><0xB5><0xA_Z>⁻₂Y + ⁴₂He + Q (Energy)
  • The atomic number decreases by 2, mass number decreases by 4.
  • Q-value (Disintegration Energy): Q = [m<0xE1><0xB5><0xA_X> - (m<0xE1><0xB5><0xA_Y> + m<0xE2><0x82><0x90>)] c² (using nuclear masses). Alpha decay occurs spontaneously only if Q > 0. This energy appears as kinetic energy of the daughter nucleus (Y) and the alpha particle.
• Beta (β) Decay: Emission of an electron (β⁻) or a positron (β⁺) from the nucleus.
  • β⁻ Decay: A neutron converts into a proton within the nucleus, emitting an electron and an antineutrino (ν̅ₑ). Occurs in neutron-rich nuclei.
    • n → p + e⁻ + ν̅ₑ
    • ᴬ<0xE1><0xB5><0xA_Z>X → ᴬ<0xE1><0xB5><0xA_Z>₊₁Y + e⁻ (or β⁻) + ν̅ₑ + Q
    • Atomic number increases by 1, mass number remains the same.
    • Q = [m<0xE1><0xB5><0xA_X> - m<0xE1><0xB5><0xA_Y>] c² (using atomic masses, as electron mass cancels out).
  • β⁺ Decay: A proton converts into a neutron within the nucleus, emitting a positron (antiparticle of electron) and a neutrino (νₑ). Occurs in proton-rich nuclei. Requires energy input if considering free proton decay, but can happen within a nucleus.
    • p → n + e⁺ + νₑ
    • ᴬ<0xE1><0xB5><0xA_Z>X → ᴬ<0xE1><0xB5><0xA_Z>⁻₁Y + e⁺ (or β⁺) + νₑ + Q
    • Atomic number decreases by 1, mass number remains the same.
    • Q = [m<0xE1><0xB5><0xA_X> - m<0xE1><0xB5><0xA_Y> - 2m<0xE1><0xB5><0x8A>] c² (using atomic masses, requires accounting for 2 electron masses).
  • Electron Capture (EC): A proton-rich nucleus captures an inner atomic electron (usually K-shell), converting a proton into a neutron and emitting a neutrino.
    • p + e⁻ → n + νₑ
    • ᴬ<0xE1><0xB5><0xA_Z>X + e⁻ → ᴬ<0xE1><0xB5><0xA_Z>⁻₁Y + νₑ + Q
    • Atomic number decreases by 1, mass number remains the same. Competes with β⁺ decay.
    • Q = [m<0xE1><0xB5><0xA_X> - m<0xE1><0xB5><0xA_Y>] c² (using atomic masses).
  • Neutrino/Antineutrino: Introduced by Pauli to explain the continuous energy spectrum of beta particles and conserve energy, linear momentum, and angular momentum in beta decay. They are chargeless, have very small (possibly zero) rest mass, and interact very weakly with matter.
• Gamma (γ) Decay: Emission of high-energy photons (gamma rays) from a nucleus in an excited state.
  • ᴬ<0xE1><0xB5><0xA_Z>X* → ᴬ<0xE1><0xB5><0xA_Z>X + γ
  • The nucleus transitions from a higher energy state to a lower energy state.
  • Often occurs after alpha or beta decay, as the daughter nucleus may be left in an excited state.
  • Neither Z nor A changes.

8. Nuclear Energy

• Energy released during nuclear reactions (fission or fusion).
• Nuclear Fission: The process of splitting a heavy, unstable nucleus (like ²³⁵U) into two or more lighter nuclei, usually triggered by neutron absorption. Releases a large amount of energy and typically 2-3 additional neutrons.
  • Example: ¹₀n + ²³⁵₉₂U → ²³⁶₉₂U* → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n + Q (≈ 200 MeV)
  • The released neutrons can cause further fission, leading to a chain reaction.
  • Controlled Chain Reaction: Used in nuclear reactors for power generation. Requires:
    • Nuclear Fuel: Fissionable material (e.g., ²³⁵U, ²³⁹Pu).
    • Moderator: Slows down fast neutrons to thermal energies, increasing the probability of fission (e.g., heavy water, graphite).
    • Control Rods: Absorb excess neutrons to control the reaction rate (e.g., Cadmium, Boron).
    • Coolant: Removes heat generated (e.g., water, molten sodium).
    • Shielding: Protects from harmful radiation.
  • Uncontrolled Chain Reaction: Basis of atomic bombs.
• Nuclear Fusion: The process of combining two or more light nuclei to form a heavier nucleus, releasing a tremendous amount of energy.
  • Example (proton-proton cycle in stars):
    • ¹₁H + ¹₁H → ²₁H + e⁺ + νₑ + 0.42 MeV
    • ²₁H + ¹₁H → ³₂He + γ + 5.49 MeV
    • ³₂He + ³₂He → ⁴₂He + ¹₁H + ¹₁H + 12.86 MeV
  • Overall: 4 ¹₁H → ⁴₂He + 2e⁺ + 2νₑ + 2γ + 26.7 MeV
  • Requires extremely high temperatures (≈ 10⁷ K) and pressures to overcome electrostatic repulsion between nuclei (thermonuclear fusion).
  • Source of energy in stars (like the Sun).
  • Basis of hydrogen bombs. Research is ongoing for controlled fusion reactors (e.g., Tokamak).

Key Conservation Laws in Nuclear Reactions:

1. Conservation of Charge (Z)
2. Conservation of Mass Number (A) (Nucleon number)
3. Conservation of Energy (including mass-energy)
4. Conservation of Linear Momentum
5. Conservation of Angular Momentum

Multiple Choice Questions (MCQs)

1. The radius R of a nucleus of mass number A is given by R = R₀ A¹ᐟ³. The value of R₀ is approximately:
   (A) 1.2 × 10⁻¹⁰ m
   (B) 1.2 × 10⁻¹³ m
   (C) 1.2 × 10⁻¹⁵ m
   (D) 1.2 × 10⁻¹⁸ m

2. Isotopes of an element have:
   (A) Same number of neutrons but different number of protons.
   (B) Same number of protons but different number of neutrons.
   (C) Same number of protons and neutrons.
   (D) Different number of protons and neutrons.

3. The binding energy per nucleon is maximum for nuclei near:
   (A) A = 2 (Helium)
   (B) A = 16 (Oxygen)
   (C) A = 56 (Iron)
   (D) A = 238 (Uranium)

4. In β⁻ decay, a nucleus ᴬ<0xE1><0xB5><0xA_Z>X transforms into Y. The representation of Y is:
   (A) ᴬ<0xE1><0xB5><0xA_Z>₊₁Y
   (B) ᴬ<0xE1><0xB5><0xA_Z>⁻₁Y
   (C) ᴬ⁻⁴<0xE1><0xB5><0xA_Z>⁻₂Y
   (D) ᴬ<0xE1><0xB5><0xA_Z>Y

5. A radioactive substance has a half-life of 30 days. The time taken for 3/4th of its original mass to disintegrate is:
   (A) 30 days
   (B) 45 days
   (C) 60 days
   (D) 75 days

6. Which of the following properties is NOT characteristic of the nuclear force?
   (A) Short range
   (B) Charge dependent
   (C) Strongest fundamental force
   (D) Saturation property

7. The energy equivalent of 1 atomic mass unit (u) is approximately:
   (A) 9.315 MeV
   (B) 93.15 MeV
   (C) 931.5 MeV
   (D) 13.6 eV

8. In nuclear reactors, Cadmium rods are used as:
   (A) Fuel
   (B) Moderator
   (C) Control rods
   (D) Coolant

9. The phenomenon of nuclear fusion requires:
   (A) Very low temperature and high pressure
   (B) Very high temperature and low pressure
   (C) Very low temperature and low pressure
   (D) Very high temperature and high pressure

10. The density of nuclear matter is:
    (A) Proportional to the mass number A
    (B) Proportional to A²
    (C) Inversely proportional to A
    (D) Nearly constant for all nuclei

Answers to MCQs:

1. (C)
2. (B)
3. (C)
4. (A)
5. (C) [Explanation: 3/4 disintegrated means 1/4 remains. Time for 1/2 to remain = T½. Time for 1/4 to remain = 2 × T½ = 2 × 30 = 60 days]
6. (B) [Nuclear force is charge independent]
7. (C)
8. (C)
9. (D)
10. (D)

Remember to thoroughly revise these concepts, especially the binding energy curve, radioactive decay laws, and the differences between fission and fusion. Practice numerical problems based on mass defect, binding energy, half-life, and activity calculations. Good luck with your preparation!