Atoms & Nuclei
Physics – MHT CET
Goal: Build a solid foundation on atomic structure, nuclear composition, binding energy, and radioactivity.
1. Simple Explanation
- Atoms are the basic units of matter.
- They are made of a nucleus (protons + neutrons) surrounded by electrons.
- The nucleus is tiny (∼10⁻¹⁵ m) but contains almost all the mass.
- Binding energy holds the nucleus together; it can be released in nuclear reactions.
- Radioactivity is the spontaneous emission of particles/energy as the nucleus seeks a lower‑energy state.
2. Key Points
| Concept | What It Means | Why It Matters |
|---|---|---|
| Atomic number (Z) | Number of protons | Determines element identity |
| Mass number (A) | Protons + neutrons | Determines isotope |
| Isotope | Same Z, different A | Different nuclear stability |
| Binding energy per nucleon | Energy needed to separate nucleus | Peak at Fe‑56; explains fusion/fission |
| Q‑value | Energy released in a reaction | Positive Q → exothermic |
| Radioactive decay modes | α, β⁻, β⁺/EC, γ, spontaneous fission | Each changes Z/A in a characteristic way |
| Half‑life (T½) | Time for half of a sample to decay | Key for radiological safety & dating |
| Fermi’s Golden Rule | Transition probability ∝ | Matrix element² × density of final states |
| Nuclear reactions | (n,γ), (p,α), (α,γ), etc. | Basis of energy generation & isotope production |
| Nuclear binding energy equation | (E_B = (Z m_p + N m_n - M_{\text{nucl}})c^2) | Quantifies stability |
| Mass defect | Difference between sum of nucleon masses & actual mass | Equal to binding energy/c² |
Binding Energy Curve
- Peaks around iron (Fe‑56).
- Fusion (lighter → heavier) releases energy until Fe.
- Fission (heavier → lighter) releases energy after Fe.
Radioactivity Overview
| Decay | Change in Z | Change in A | Typical Emitted Particle | Example |
|---|---|---|---|---|
| α (α‑decay) | −2 | −4 | ²⁴He nucleus | ²³⁸U → ²³⁴Th + α |
| β⁻ | +1 | +0 | e⁻ + ν̄_e | ¹³⁶Xe → ¹³⁶Cs + β⁻ |
| β⁺ / EC | −1 | +0 | e⁺ + ν_e (EC) | ¹⁰⁰Mo → ¹⁰⁰Nb + β⁺ |
| γ | 0 | 0 | γ‑ray | ²³⁶U* → ²³⁶U + γ |
| Spontaneous fission | variable | variable | two fragments + neutrons | ²³⁵U → ¹⁴⁰Xe + ⁹⁵Sr + 2n |
3. Worked Example
Problem
Calculate the binding energy per nucleon for the nucleus of ⁴He (helium‑4).
Given
- Mass of a proton (m_p = 1.007276,\text{u})
- Mass of a neutron (m_n = 1.008665,\text{u})
- Mass of ⁴He nucleus (M_{\text{nucl}} = 4.002603,\text{u})
- 1 atomic mass unit (u) = (931.5,\text{MeV}/c^2)
Solution
-
Compute the sum of individual nucleon masses
[ M_{\text{sum}} = 2m_p + 2m_n = 2(1.007276) + 2(1.008665) = 4.031882,\text{u} ] -
Find the mass defect
[ \Delta m = M_{\text{sum}} - M_{\text{nucl}} = 4.031882 - 4.002603 = 0.029279,\text{u} ] -
Convert mass defect to energy
[ E_B = \Delta m \times 931.5,\frac{\text{MeV}}{c^2} = 0.029279 \times 931.5 \approx 27.3,\text{MeV} ] -
Binding energy per nucleon
[ \frac{E_B}{A} = \frac{27.3,\text{MeV}}{4} \approx 6.8,\text{MeV/nucleon} ]
Answer: The binding energy per nucleon for ⁴He is ≈ 6.8 MeV.
Interpretation: ⁴He is very stable; the high binding energy per nucleon explains why helium is produced in stellar nucleosynthesis and why heavier nuclei undergo fusion to release energy.
4. Quick Revision Summary
- Atom = nucleus + electrons.
- Nucleus: Protons (Z) + neutrons (N).
- Isotope: Same Z, different A.
- Binding energy = mass defect × (c^2).
- Peak binding energy at Fe‑56 → explains fusion/fission energetics.
- Decay modes: α, β⁻, β⁺/EC, γ, spontaneous fission.
- Q‑value (energy released) = initial mass – final mass.
- Half‑life expresses radioactivity rate.
- Nuclear reactions: (n,γ), (p,α), (α,γ), etc. used in power & medicine.
- Mass defect ⇒ binding energy per nucleon.
Tip for CET: Memorize common isotopes