Atoms and Nuclei

1. Introduction

Atomic physics explores the structure of atoms and nuclei. From Rutherford's nuclear model to Bohr's quantum theory, this chapter traces our understanding of the microscopic world.

2. Rutherford's Model

2.1 Alpha Particle Scattering Experiment

Most alpha particles passed through thin gold foil undeflected, some were deflected, and very few bounced back.

2.2 Conclusions

• Most of the atom is empty space.
• The positive charge and most of the mass are concentrated in a tiny nucleus.
• The nucleus is about 10^{-15} m in diameter (atom: 10^{-10} m).

2.3 Limitations

Could not explain the stability of atoms (accelerating electrons should radiate energy and spiral into the nucleus) or the discrete spectral lines.

3. Bohr's Model

3.1 Postulates

1. Electrons move in circular orbits (stationary states) without radiating energy.
2. Angular momentum is quantized: mvr = nh/2π.
3. Electrons can jump between orbits, emitting/absorbing photons: ΔE = hf.

3.2 Energy Levels

E_n = -13.6/n² eV (for hydrogen). Radius: r_n = n²a₀, where a₀ = 0.529 angstroms (Bohr radius). Velocity: v_n = v₁/n, where v₁ = c/137.

3.3 Limitations

Works only for single-electron atoms. Cannot explain fine structure or intensity of spectral lines.

4. Hydrogen Spectrum

The hydrogen spectrum consists of several spectral series:

Lyman series (n₁ = 1): Ultraviolet. 1/λ = R(1 - 1/n²) Balmer series (n₁ = 2): Visible. 1/λ = R(1/4 - 1/n²) Paschen series (n₁ = 3): Infrared. Brackett series (n₁ = 4): Infrared. Pfund series (n₁ = 5): Infrared.

R = 1.097×10⁷ m^{-1} is the Rydberg constant.

5. The Nucleus

5.1 Composition

Nucleus contains protons (positive) and neutrons (neutral). Mass number A = Z + N.

5.2 Size

Nuclear radius: R = R₀A^{1/3}, where R₀ = 1.2 × 10^{-15} m.

5.3 Nuclear Density

Extremely high — about 10¹⁷ kg/m³, independent of A.

6. Radioactivity

6.1 Types of Decay

Alpha decay: Z^A X → Z-2^{A-4} Y + ₂⁴He. Decreases A by 4, Z by 2. Beta decay: n → p + e⁻ + ν̄. Neutron converts to proton. Gamma decay: High-energy photon emission from excited nucleus.

6.2 Law of Radioactive Decay

N = N₀ e^{-λt}, where λ is the decay constant. Half-life: T_{1/2} = ln 2/λ = 0.693/λ. Mean life: τ = 1/λ = T_{1/2}/0.693.

6.3 Activity

A = λN = A₀ e^{-λt}. Unit: becquerel (Bq) = 1 decay/s.

7. Nuclear Fission and Fusion

Fission: Heavy nucleus splits into lighter nuclei, releasing energy. Example: ₉₂²³⁵U + n → ₅₆¹⁴¹Ba + ₃₆⁹²Kr + 3n + energy.

Chain reaction: Controlled (nuclear reactor) vs uncontrolled (atomic bomb).

Fusion: Light nuclei combine to form a heavier nucleus. Example: ₁²H + ₁³H → ₂⁴He + n + 17.6 MeV.

Fusion requires very high temperature (10⁷-10⁸ K) to overcome Coulomb repulsion.

8. Worked Problems

Problem 1: Find the energy of the electron in the second excited state of hydrogen (n = 3). Solution: E₃ = -13.6/3² = -13.6/9 = -1.51 eV.

Problem 2: A radioactive sample has half-life of 10 days. What fraction remains after 30 days? Solution: N/N₀ = (1/2)^{t/T} = (1/2)^{30/10} = (1/2)³ = 1/8.

Problem 3: Calculate the wavelength of Hα line (Balmer series, n = 3 to n = 2). Solution: 1/λ = R(1/4 - 1/9) = R(5/36) = 1.097×10⁷×5/36 = 1.524×10⁶. λ = 6.56×10^{-7} m = 656 nm.

9. Common Mistakes

'Students often confuse atomic number Z (protons) with mass number A (protons + neutrons). Isotopes have same Z, different A.'

'In the Bohr model, energy is negative — this represents the binding energy of the electron. An electron with E = 0 is free.'

10. ISC Exam Focus

11. Self-Test Questions

1. State the postulates of Bohr's model of the atom. Derive the expression for the radius of the nth orbit.
2. Calculate the wavelength of the first line of the Lyman series (R = 1.097×10⁷ m^{-1}).
3. A radioactive source has an activity of 10⁶ Bq and half-life of 2 years. Find its activity after 6 years.
4. Distinguish between nuclear fission and nuclear fusion with examples.
5. The decay constant of a substance is 0.0231 year^{-1}. Find its half-life and mean life.