class 12 notes

Class 12 Physics Notes Chapter 5 (Nuclei) – Physics Part-II Book

Philoid

Philoid

— 9 min read

Physics Part-II
Alright class, let's get straight into Chapter 5: Nuclei. This is a crucial chapter, not just for your board exams but also for various government competitive exams where physics is a component. Pay close attention to the definitions, formulas, and concepts.

Chapter 5: Nuclei - Detailed Notes for Government Exam Preparation

1. Introduction & Basic Terminology

  • Nucleus: The central core of an atom, discovered by Rutherford's alpha-scattering experiment. It contains protons and neutrons, collectively called nucleons.
  • Atomic Number (Z): Number of protons in the nucleus. It determines the chemical element.
  • Mass Number (A): Total number of protons and neutrons in the nucleus (A = Z + N, where N is the neutron number).
  • Nuclide: A specific type of nucleus characterized by its atomic number (Z) and mass number (A). Represented as ᴬ<0xE1><0xB5><0xA3>X, where X is the chemical symbol.
  • Atomic Mass Unit (amu or u): Defined as 1/12th the mass of an unexcited Carbon-12 atom.
    • 1 amu = 1.660539 × 10⁻²⁷ kg ≈ 931.5 MeV/c² (This energy equivalence is vital!)
  • Isotopes: Nuclides having the same atomic number (Z) but different mass numbers (A). They have the same chemical properties but different physical properties (e.g., ¹¹H, ²¹H (Deuterium), ³¹H (Tritium) are isotopes of Hydrogen).
  • Isobars: Nuclides having the same mass number (A) but different atomic numbers (Z). (e.g., ³¹H and ³₂He).
  • Isotones: Nuclides having the same number of neutrons (N = A - Z). (e.g., ³¹H and ⁴₂He, both have N=2).

2. Size and Density of the Nucleus

  • Nuclear Radius (R): Experimentally found to be proportional to the cube root of the mass number (A).
    • Formula: R = R₀ A¹ᐟ³
    • Where R₀ (empirical constant) ≈ 1.2 × 10⁻¹⁵ m = 1.2 femtometre (fm). (Value can slightly vary, 1.1 fm to 1.3 fm is common range).
  • Nuclear Volume (V): 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): Density = Mass / Volume.
    • Mass ≈ A × (mass of one nucleon ≈ mₚ ≈ m<0xE2><0x82><0x99>)
    • ρ ≈ (A × mₚ) / [(4/3)πR₀³ A] = 3mₚ / (4πR₀³)
    • Key Point: Nuclear density is approximately constant for all nuclei and is extremely high (≈ 2.3 × 10¹⁷ kg/m³). It is independent of the mass number A.

3. Mass-Energy Equivalence & Nuclear Binding Energy

  • Einstein's Mass-Energy Relation: E = mc²
    • Mass and energy are inter-convertible. This is fundamental to understanding nuclear reactions.
    • 1 amu = 931.5 MeV/c² implies that converting 1 amu of mass completely into energy yields 931.5 MeV.
  • Mass Defect (Δm): The difference between the sum of the masses of constituent nucleons (protons and neutrons) in their free state and the actual mass of the nucleus.
    • Mass of nucleus (M<0xE2><0x82><0x99><0xE1><0xB5><0xA3><0xE1><0xB5><0x84>) is always less than the sum of masses of its constituents.
    • Formula: Δm = [ Z mₚ + (A - Z) m<0xE2><0x82><0x99> ] - M<0xE2><0x82><0x99><0xE1><0xB5><0xA3><0xE1><0xB5><0x84>
      • mₚ = mass of proton, m<0xE2><0x82><0x99> = mass of neutron.
      • Often, calculations use atomic masses instead of nuclear masses. In that case, use the mass of the Hydrogen atom (m<0xE1><0xB5><0x8B>) and subtract the mass of electrons:
        Δm = [ Z m<0xE1><0xB5><0x8B> + (A - Z) m<0xE2><0x82><0x99> ] - M<0xE2><0x82><0x99><0xE1><0xB5><0x8A><0xE1><0xB5><0x9C><0xE1><0xB5><0x90><0xE1><0xB5><0x8B> (where M<0xE2><0x82><0x99><0xE1><0xB5><0x8A><0xE1><0xB5><0x9C><0xE1><0xB5><0x90><0xE1><0xB5><0x8B> is the atomic mass of the nuclide ᴬ<0xE1><0xB5><0xA3>X).
  • Nuclear Binding Energy (BE or E<0xE1><0xB5><0x87>): The energy equivalent to the mass defect. It represents the energy required to completely separate all the nucleons from the nucleus to an infinite distance. It's also the energy released when the constituent nucleons combine to form the nucleus.
    • Formula: BE = Δm c² = [ { Z mₚ + (A - Z) m<0xE2><0x82><0x99> } - M<0xE2><0x82><0x99><0xE1><0xB5><0xA3><0xE1><0xB5><0x84> ] c²
    • Practical Formula: BE (in MeV) = Δm (in amu) × 931.5 MeV/amu
  • Binding Energy per Nucleon (BE/A or E<0xE1><0xB5><0x87><0xE2><0x82><0x99>): The average energy required to remove one nucleon from the nucleus. BE/A = BE / A.
    • Significance: BE/A is a measure of the stability of the nucleus. Higher BE/A means a more stable nucleus.
    • Binding Energy Curve: A plot of BE/A versus mass number (A).
      • Features:
        • Low BE/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 BE/A ≈ 8.75 MeV. These are the most stable nuclei.
        • Gradual decrease for heavy nuclei (A > 170).
      • Inferences:
        • Fusion: Light nuclei can fuse to form heavier nuclei (up to A≈56), increasing BE/A and releasing energy.
        • Fission: Heavy nuclei can split into lighter nuclei, increasing BE/A and releasing energy.

4. Nuclear Force

  • The strong force that binds protons and neutrons together within the tiny nucleus, overcoming the immense electrostatic repulsion between protons.
  • Properties:
    • Strongest force: Much stronger than electromagnetic and gravitational forces within the nucleus.
    • Short-range: Acts only over very short distances (≈ few fm, typically 2-3 fm). Negligible outside the nucleus.
    • 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 is why BE/A is 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.

5. Radioactivity

  • The spontaneous disintegration of unstable nuclei by emitting particles (alpha, beta) or electromagnetic radiation (gamma rays). Discovered by Henri Becquerel.

  • 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><0xA3>X → ᴬ⁻⁴<0xE1><0xB5><0xA3>⁻₂Y + ⁴₂He + Q (Q is disintegration energy)
      • Z decreases by 2, A decreases by 4.
      • α-particles: +2e charge, relatively heavy, low penetration power, high ionization power.
    • Beta (β) Decay: Emission of an electron (β⁻) or a positron (β⁺).
      • β⁻ Decay: A neutron converts into a proton, emitting an electron and an antineutrino (ν̅). Occurs in neutron-rich nuclei.
        • n → p + e⁻ + ν̅
        • ᴬ<0xE1><0xB5><0xA3>X → ᴬ<0xE1><0xB5><0xA3>₊₁Y + e⁻ + ν̅ + Q
        • Z increases by 1, A remains unchanged.
      • β⁺ Decay: A proton converts into a neutron, emitting a positron and a neutrino (ν). Occurs in proton-rich nuclei. Requires energy input or occurs within the nucleus where binding energy changes compensate.
        • p → n + e⁺ + ν
        • ᴬ<0xE1><0xB5><0xA3>X → ᴬ<0xE1><0xB5><0xA3>⁻₁Y + e⁺ + ν + Q
        • Z decreases by 1, A remains unchanged.
      • β-particles (e⁻ or e⁺): -e or +e charge, light mass, moderate penetration power, moderate ionization power. Neutrinos/antineutrinos are massless (or nearly massless), chargeless particles interacting very weakly with matter.
    • Gamma (γ) Decay: Emission of high-energy photons (gamma rays). Occurs when a nucleus is in an excited state (often after α or β decay) and transitions to a lower energy state.
      • ᴬ<0xE1><0xB5><0xA3>X* → ᴬ<0xE1><0xB5><0xA3>X + γ (* denotes excited state)
      • Z and A remain unchanged.
      • γ-rays: No charge, massless photons, high penetration power, low ionization power.

  • Law of Radioactive Decay: The rate of disintegration (-dN/dt) is directly proportional to the number of radioactive nuclei (N) present at that instant.

    • Formula: dN/dt = -λN
    • Integrating, we get: N(t) = N₀ e⁻<0xE2><0x81><0xBB>ᵗ
      • N(t) = Number of undecayed nuclei at time t.
      • N₀ = Initial number of nuclei at t = 0.
      • λ = Decay constant or disintegration constant (unit: s⁻¹, min⁻¹, year⁻¹, etc.). It's characteristic of the radioactive substance.

  • Activity (R): The rate of decay, i.e., the number of disintegrations per second.

    • Formula: R(t) = -dN/dt = λN(t) = λN₀ e⁻<0xE2><0x81><0xBB>ᵗ = R₀ e⁻<0xE2><0x81><0xBB>ᵗ
    • R₀ = λN₀ is the initial activity.
    • Units: Becquerel (Bq) = 1 disintegration/second (SI unit). Curie (Ci) = 3.7 × 10¹⁰ Bq (older unit). Rutherford (Rd) = 10⁶ Bq.

  • Half-Life (T₁/₂): The time taken for the number of radioactive nuclei (or activity) to reduce to half its initial value.

    • At t = T₁/₂, N(t) = N₀/2.
    • N₀/2 = N₀ e⁻<0xE2><0x81><0xBB>ᵀ₁/₂ => e<0xE2><0x81><0xBB>ᵀ₁/₂ = 2
    • Formula: T₁/₂ = ln(2) / λ = 0.693 / λ
    • After n half-lives, N = N₀ / 2ⁿ.

  • Mean Life or Average Life (τ): The average lifetime of all the radioactive nuclei in a sample.

    • Formula: τ = 1 / λ
    • Relation with Half-Life: τ = T₁/₂ / ln(2) = T₁/₂ / 0.693 ≈ 1.44 T₁/₂

6. Nuclear Energy

  • Energy released during nuclear reactions (fission or fusion) due to the conversion of mass into energy according to E = mc².
  • Nuclear Fission: The process of splitting a heavy nucleus (like Uranium-235) into two or more lighter nuclei, usually triggered by neutron bombardment. A large amount of energy is released along with more neutrons.
    • Example: ¹₀n + ²³⁵<0xE2><0x82><0x92>U → ²³⁶<0xE2><0x82><0x92>U* → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n + Q (Q ≈ 200 MeV)
    • Chain Reaction: The neutrons released in fission can cause further fissions, leading to a chain reaction.
      • Controlled: Used in nuclear reactors (maintain multiplication factor K ≈ 1).
      • Uncontrolled: Used in atomic bombs (K > 1).
    • Nuclear Reactor: A device to control nuclear fission for power generation. Key components:
      • Fuel: Fissionable material (e.g., U-235, Pu-239).
      • Moderator: Slows down fast neutrons (e.g., heavy water D₂O, graphite). Slow neutrons (thermal neutrons) are more effective in causing fission of U-235.
      • Control Rods: Absorb excess neutrons to control the reaction rate (e.g., Cadmium, Boron).
      • Coolant: Removes heat generated (e.g., water, heavy water, liquid sodium).
      • Shielding: Protects from harmful radiation (e.g., thick concrete walls).
  • 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⁺ + ν + 0.42 MeV
      • ²¹H + ¹¹H → ³₂He + γ + 5.49 MeV
      • ³₂He + ³₂He → ⁴₂He + ¹¹H + ¹¹H + 12.86 MeV
    • Conditions: Requires extremely high temperature (≈ 10⁷ K) and pressure to overcome electrostatic repulsion between nuclei (Thermonuclear reactions).
    • Energy Source: Sun and stars generate energy through fusion. Hydrogen bomb uses fusion.
    • Advantages over Fission: Fuel (like Deuterium) is abundant, produces less radioactive waste, potentially higher energy yield per unit mass. Controlling fusion for power generation is a major ongoing research area (e.g., Tokamak).

Multiple Choice Questions (MCQs)

  1. The radius (R) of a nucleus is related to its mass number (A) by the approximate relation:
    (a) R ∝ A¹/²
    (b) R ∝ A
    (c) R ∝ A¹/³
    (d) R ∝ A²/³

  2. Two nuclei have mass numbers in the ratio 1:8. What is the ratio of their nuclear radii?
    (a) 1:2
    (b) 2:1
    (c) 1:4
    (d) 4:1

  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 a β⁻ decay process, inside the nucleus:
    (a) A proton converts into a neutron, electron, and neutrino.
    (b) A neutron converts into a proton, electron, and antineutrino.
    (c) A proton converts into a neutron, positron, and antineutrino.
    (d) A neutron converts into a proton, positron, and neutrino.

  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 particles has the highest penetration power?
    (a) Alpha particle
    (b) Beta particle
    (c) Gamma ray
    (d) Proton

  7. The function of a moderator in a nuclear reactor is to:
    (a) Absorb neutrons
    (b) Accelerate neutrons
    (c) Slow down fast neutrons
    (d) Remove heat from the core

  8. Nuclear forces are:
    (a) Short-range and charge-dependent
    (b) Long-range and charge-independent
    (c) Short-range and charge-independent
    (d) Long-range and charge-dependent

  9. The energy equivalent of 1 atomic mass unit (amu) is approximately:
    (a) 1.6 × 10⁻¹⁹ J
    (b) 9.1 × 10⁻³¹ J
    (c) 931.5 MeV
    (d) 1 MeV

  10. Nuclear fusion requires extremely high temperatures because:
    (a) The nuclei need high kinetic energy to overcome electrostatic repulsion.
    (b) The mass defect is only significant at high temperatures.
    (c) Neutrons are only produced at high temperatures.
    (d) The nuclear force is only active at high temperatures.

Answer Key for MCQs:

  1. (c)
  2. (a) [R₁/R₂ = (A₁/A₂)¹/³ = (1/8)¹/³ = 1/2]
  3. (c)
  4. (b)
  5. (c) [If 3/4 disintegrates, 1/4 remains. N = N₀/4 = N₀/(2²). This takes 2 half-lives. Time = 2 * T₁/₂ = 2 * 30 = 60 days]
  6. (c)
  7. (c)
  8. (c)
  9. (c)
  10. (a)

Study these notes thoroughly. Focus on understanding the concepts behind the formulas, especially the binding energy curve and the laws of radioactive decay. These are frequently tested areas. Let me know if any part needs further clarification.

Read more