Atoms
'The atom is mostly EMPTY space — a tiny nucleus surrounded by electrons. But how do the electrons stay in place? Quantum mechanics has the answer.'
1. Chapter Overview
This chapter traces the EVOLUTION of our understanding of atomic structure. Topics include: RUTHERFORD'S ALPHA-PARTICLE SCATTERING EXPERIMENT (which discovered the nucleus), the LIMITATIONS of the Rutherford model, BOHR'S MODEL of the hydrogen atom (combining classical and quantum ideas), the ENERGY LEVELS of hydrogen, the HYDROGEN SPECTRUM (Lyman, Balmer, Paschen series), and the LINE SPECTRA of atoms.
2. Rutherford's Model of the Atom (1911)
The Alpha-Particle Scattering Experiment
- Setup: Alpha particles (He²⁺) were fired at a thin gold foil. A fluorescent screen detected their positions.
- Observations:
- MOST alpha particles passed through UNDEFLECTED.
- SOME were deflected by SMALL angles.
- A VERY FEW (about 1 in 8000) were deflected by MORE than 90° — some even BOUNCED BACK.
Conclusions (Rutherford's Nuclear Model)
- Most of the atom is EMPTY SPACE — explains why most alpha particles pass through.
- The atom has a TINY, DENSE, POSITIVELY CHARGED nucleus — explains the large-angle deflections.
- The positive charge and almost ALL the mass of the atom are concentrated in the nucleus.
- Electrons REVOLVE around the nucleus in circular orbits — like planets around the Sun.
Limitations of Rutherford's Model
- Instability: According to Maxwell's equations, ACCELERATING electrons (moving in a circle = centripetal acceleration) should RADIATE ENERGY. The electron would spiral into the nucleus in about 10⁻¹¹ seconds — atoms should NOT exist!
- Line spectra: Rutherford's model cannot explain why atoms emit only DISCRETE WAVELENGTHS of light (line spectra).
3. Bohr's Model of the Hydrogen Atom (1913)
Bohr's Postulates
- Stationary orbits: Electrons can exist ONLY in certain PERMITTED circular orbits (stationary states). In these orbits, they do NOT radiate energy.
- Angular momentum quantisation: mvr = nh/(2π), where n = 1, 2, 3, ... (PRINCIPAL QUANTUM NUMBER).
- Energy transitions: An electron can jump from one orbit to another by ABSORBING or EMITTING a photon of energy hf = Eᵢ − E_f.
4. Energy and Radius of Bohr Orbits
Radius
- r_n = n²(ε₀h²)/(πme²) = n² × a₀, where a₀ = 0.529 × 10⁻¹⁰ m (BOHR RADIUS).
- r_n ∝ n² — 'The radius increases as the SQUARE of the principal quantum number.'
Velocity
- v_n = e²/(2ε₀hn) — v_n ∝ 1/n.
Energy
- E_n = −13.6/n² eV (for hydrogen). E₁ = −13.6 eV (ground state), E₂ = −3.4 eV, E₃ = −1.51 eV, E₄ = −0.85 eV.
- 'The energy is NEGATIVE — the electron is bound to the nucleus. To FREE the electron (ionise), we need to give it +13.6 eV of energy.'
Worked Example 1
Problem: Find the radius of the second Bohr orbit in hydrogen. Solution: r₂ = n²a₀ = 4 × 0.529 × 10⁻¹⁰ = 2.116 × 10⁻¹⁰ m = 2.116 Å.
5. Hydrogen Spectrum
Energy Level Transitions
- hf = Eᵢ − E_f = 13.6(1/n_f² − 1/nᵢ²) eV.
Spectral Series
| Series | n_f | nᵢ | Region | Formula |
|---|---|---|---|---|
| Lyman | 1 | 2, 3, 4, ... | ULTRAVIOLET | 1/λ = R(1 − 1/n²) |
| Balmer | 2 | 3, 4, 5, ... | VISIBLE | 1/λ = R(1/4 − 1/n²) |
| Paschen | 3 | 4, 5, 6, ... | INFRARED | 1/λ = R(1/9 − 1/n²) |
| Brackett | 4 | 5, 6, 7, ... | INFRARED | 1/λ = R(1/16 − 1/n²) |
| Pfund | 5 | 6, 7, 8, ... | INFRARED | 1/λ = R(1/25 − 1/n²) |
- R = 1.097 × 10⁷ m⁻¹ (Rydberg constant). The Rydberg formula: 1/λ = R(1/n_f² − 1/nᵢ²).
Worked Example 2
Problem: Find the wavelength of the first line of the Balmer series (n=3 → n=2). Solution: 1/λ = R(1/4 − 1/9) = R(5/36) = 1.097×10⁷×5/36 = 1.524×10⁶ m⁻¹. λ = 1/1.524×10⁶ = 6.56×10⁻⁷ m = 656 nm (RED line of hydrogen).
6. Comparison Table: Rutherford vs Bohr Model
| Feature | Rutherford Model | Bohr Model |
|---|---|---|
| Nucleus | Small, dense, positive | Small, dense, positive |
| Electron orbits | Any orbit allowed | ONLY certain QUANTISED orbits |
| Radiation | Should radiate continuously | NO radiation in stationary states |
| Angular momentum | Not quantised | mvr = nh/(2π) |
| Spectrum | Cannot explain line spectra | EXPLAINS hydrogen spectrum precisely |
| Stability | Unstable | STABLE (by postulate) |
| Validity | FAILS for atoms | Works for hydrogen-like atoms |
7. Common Mistakes
- Bohr radius formula: a₀ = ε₀h²/(πme²) = 0.529 × 10⁻¹⁰ m. Many students forget this is the FIRST orbit radius.
- Energy levels are NEGATIVE: E_n = −13.6/n² eV. The negative sign indicates the electron is BOUND. Ionisation energy is +13.6 eV for hydrogen.
- Lyman vs Balmer: Lyman series (n_f = 1) is in UV, Balmer series (n_f = 2) is in VISIBLE. 'Remember: the Balmer lines are the ones we can see.'
- Rydberg constant units: R = 1.097×10⁷ m⁻¹. The formula gives 1/λ, not λ directly.
8. CBSE Exam Focus
- Rutherford's experiment — observations, conclusions, limitations
- Bohr's postulates — stationary orbits, angular momentum quantisation, energy transitions
- Bohr radius and energy — r_n = n²a₀, E_n = −13.6/n² eV
- Hydrogen spectrum — Lyman, Balmer, Paschen series
- Rydberg formula — 1/λ = R(1/n_f² − 1/nᵢ²)
- Energy level diagram — transitions, ionisation energy
9. Self-Test
Q1: Find the energy required to excite a hydrogen atom from n=1 to n=3. A1: E₁ = −13.6 eV, E₃ = −13.6/9 = −1.51 eV. ΔE = E₃ − E₁ = −1.51 − (−13.6) = 12.09 eV.
Q2: Find the wavelength of the photon emitted when an electron falls from n=4 to n=2 in hydrogen. A2: 1/λ = R(1/4 − 1/16) = R(3/16) = 1.097×10⁷×3/16 = 2.057×10⁶ m⁻¹. λ = 4.86×10⁻⁷ m = 486 nm.
Q3: Calculate the ionisation energy of hydrogen in eV and J. A3: IE = 0 − E₁ = 0 − (−13.6) = 13.6 eV = 13.6×1.6×10⁻¹⁹ = 2.176×10⁻¹⁸ J.
Q4: What is the angular momentum of an electron in the third Bohr orbit? A4: L = nh/(2π) = 3×6.63×10⁻³⁴/(2π) = 19.89×10⁻³⁴/6.28 = 3.17×10⁻³⁴ kg·m²/s.
Q5: In Rutherford's experiment, why did only a FEW alpha particles bounce back? A5: Because the nucleus is VERY SMALL and concentrated. Only alpha particles heading DIRECTLY towards a nucleus experienced a strong repulsive force. Most passed through the empty space.
10. Conclusion
The study of atoms REVEALED the quantum world:
- RUTHERFORD: 'Discovered the nucleus — the atom's dense core. But his model could not explain stability or spectra.'
- BOHR: 'Introduced QUANTISATION — electrons exist only in specific orbits with specific energies. The hydrogen spectrum was EXPLAINED.'
- SPECTRUM: 'Every atom has a UNIQUE spectral fingerprint — the basis of spectroscopy and astrophysics.'
- 'Bohr's model was a BRIDGE between classical physics and true quantum mechanics — it was NOT the final answer, but it was a GIANT step forward.'
'Bohr's model gave us the quantum atom — quantised orbits, energy levels, and the explanation of spectral lines.'
