CBSE Class 12 Physics Chapter 13 Nuclei Notes

Key Nuclear Terminology

Nucleons

Collective term for protons and neutrons residing within the atomic nucleus.

Atomic Number (Z)

The number of protons in an atom's nucleus, determining its chemical element.

Mass Number (A)

The total number of protons and neutrons (nucleons) in an atom's nucleus. A = Z + N.

Isotopes

Atoms of the same element (same Z) but with different numbers of neutrons (different A).

Isobars

Atoms with the same mass number (A) but different atomic numbers (Z).

Isotones

Atoms with the same number of neutrons (N) but different atomic numbers (Z) and mass numbers (A).

Mass Defect (Δm)

The difference between the sum of the masses of individual nucleons and the actual measured mass of the nucleus. Δm = [Zmp + (A-Z)mn] - M(nucleus).

Binding Energy (EB)

The energy required to break a nucleus into its constituent protons and neutrons. EB = Δm c².

Radioactivity

The spontaneous emission of radiation (alpha, beta, gamma) from unstable atomic nuclei to achieve a more stable configuration.

Nuclear Forces and Stability

The stability of an atomic nucleus is a fascinating interplay of forces. Inside the tiny nucleus, protons repel each other strongly due to their positive electric charge (Coulomb force). This repulsive force is immense given the extremely small distances. However, the nucleus doesn't fly apart; it's held together by an even stronger attractive force known as the strong nuclear force or nuclear force.

Key characteristics of the nuclear force:

• It is the strongest fundamental force in nature, much stronger than electromagnetic or gravitational forces.
• It is short-ranged, acting only over distances of about 10⁻¹⁵ m (femto-meters). Beyond this range, its strength rapidly drops to zero.
• It is charge-independent, meaning it acts equally between proton-proton, neutron-neutron, and proton-neutron pairs.
• It is spin-dependent, slightly stronger when nucleon spins are parallel.
• It exhibits saturation property, meaning a nucleon interacts only with its immediate neighbours, not all nucleons in the nucleus.

Nuclear stability is often assessed by the binding energy per nucleon (EB/A). A higher binding energy per nucleon indicates a more stable nucleus. The plot of binding energy per nucleon versus mass number (A) shows a peak around A = 56 (Iron), indicating that iron is the most stable element. Nuclei lighter than iron tend to undergo fusion to become more stable, while nuclei heavier than iron tend to undergo fission to achieve stability. This curve explains why energy is released during both nuclear fission and fusion reactions.

Must Remember: Key Formulas & Concepts

• Nuclear Radius (R): R = R₀ A¹/³, where R₀ ≈ 1.2 x 10⁻¹⁵ m (1.2 fm) and A is the mass number.
• Nuclear Density: Is approximately constant for all nuclei (ρ ≈ 2.3 x 10¹⁷ kg/m³).
• Einstein's Mass-Energy Equivalence: E = mc², where c is the speed of light (3 x 10⁸ m/s). 1 amu = 931.5 MeV/c².
• Binding Energy (EB): EB = Δm c² = ([Zmp + (A-Z)mn] - M(nucleus))c².
• Radioactive Decay Law: N(t) = N₀ e⁻λt, where N₀ is initial nuclei, N(t) is nuclei at time t, λ is decay constant.
• Half-Life (T₁/₂): Time taken for half of the radioactive nuclei to decay. T₁/₂ = ln(2) / λ = 0.693 / λ.
• Mean Life (τ): Average lifetime of a radioactive nucleus. τ = 1 / λ = T₁/₂ / 0.693 ≈ 1.44 T₁/₂.
• Activity (R): Rate of decay. R = -dN/dt = λN = R₀ e⁻λt. Unit: Becquerel (Bq) or Curie (Ci).
• Alpha Decay: Nucleus emits an alpha particle (₂⁴He). Atomic number decreases by 2, mass number by 4.
• Beta Decay (β⁻): Neutron converts to proton + electron + antineutrino. Z increases by 1, A remains same.
• Beta Decay (β⁺): Proton converts to neutron + positron + neutrino. Z decreases by 1, A remains same.
• Gamma Decay: Nucleus in excited state emits a gamma ray photon (high-energy photon) to de-excite. No change in Z or A.

Worked Examples

• {"title":"Example 1: Half-Life Calculation","bodyMarkdown":"Q: A radioactive substance has a half-life of 10 days. If you start with 1000 atoms, how many atoms will remain after 30 days?\nA: \n1. Number of half-lives = Total time / Half-life = 30 days / 10 days = 3.\n2. After 1 half-life: 1000 / 2 = 500 atoms.\n3. After 2 half-lives: 500 / 2 = 250 atoms.\n4. After 3 half-lives: 250 / 2 = 125 atoms.\nAlternatively, N(t) = N₀ (1/2)^(t/T₁/₂) = 1000 (1/2)³ = 1000 * (1/8) = 125 atoms."}
• {"title":"Example 2: Binding Energy Conversion","bodyMarkdown":"Q: Calculate the energy equivalent of 1 atomic mass unit (amu) in MeV.\nA: \n1. 1 amu = 1.6605 x 10⁻²⁷ kg.\n2. Using E = mc²: E = (1.6605 x 10⁻²⁷ kg) * (3 x 10⁸ m/s)²\n3. E = 1.49245 x 10⁻¹⁰ J.\n4. Convert Joules to eV: 1 eV = 1.602 x 10⁻¹⁹ J. So, E = (1.49245 x 10⁻¹⁰ J) / (1.602 x 10⁻¹⁹ J/eV) = 9.316 x 10⁸ eV.\n5. E ≈ 931.5 MeV. (This is a standard conversion factor you should remember!)"}

Nuclear Fission vs. Nuclear Fusion

Board Exam Trap: Radioactivity Calculations

Students often confuse half-life (T₁/₂) and mean life (τ). Remember, half-life is the time for half the sample to decay, while mean life is the average lifetime of a nucleus. Also, be careful with units: ensure consistency (e.g., time in seconds, decay constant in s⁻¹). When calculating binding energy, remember to use atomic masses and convert the mass defect into energy using the 931.5 MeV/amu conversion factor. Clearly state the decay equations for alpha, beta, and gamma emissions, showing conservation of mass number and atomic number. Practice drawing the binding energy per nucleon curve and explaining its significance for fission and fusion.

Practice Questions with Solutions

• Q1: What are the main characteristics of nuclear forces? A1: Strongest, short-ranged, charge-independent, spin-dependent, and exhibit saturation property.
• Q2: A nucleus undergoes beta-minus decay. How do its atomic number and mass number change? A2: Atomic number (Z) increases by 1, and mass number (A) remains unchanged.
• Q3: Why is energy released during both nuclear fission and fusion? A3: Both processes lead to products with higher binding energy per nucleon (more stable), meaning some mass is converted into energy (mass defect).
• Q4: What is the significance of the constant nuclear density? A4: It implies that the volume of a nucleus is directly proportional to its mass number (V ∝ A), suggesting nucleons pack together with roughly the same average spacing regardless of nucleus size.

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