The Atoms chapter traces how the alpha-particle scattering experiment of Geiger and Marsden led Rutherford to the nuclear model, and how Bohr then added quantum postulates to explain the stable orbits and line spectrum of hydrogen. It covers the distance of closest approach, the radius, speed and energy of Bohr orbits, the hydrogen spectral series through the Rydberg formula, and de Broglie’s wave explanation of quantisation. It is a high-yield NEET topic because nearly every year asks numerical questions on Bohr-orbit formulas and the hydrogen spectrum.
Key Concepts
1. Rutherford’s Nuclear Model
Alpha particle scattering experiment showed that the atom has a tiny, dense, positively charged nucleus at the centre, with electrons orbiting around it.
Limitation: An orbiting electron should continuously radiate energy and spiral into the nucleus - but atoms are stable. Classical physics couldn’t explain this.
2. Bohr’s Model of Hydrogen Atom
Postulates
- Electrons revolve in fixed circular orbits (stationary orbits) without radiating energy
- Quantisation: Angular momentum is quantised: L = mvr = nh/(2π), n = 1, 2, 3…
- Energy is emitted/absorbed only when an electron jumps between orbits: E = hν = E_i − E_f
Key Formulae for Hydrogen (Z = 1)
| Quantity | Formula |
|---|---|
| Radius of nth orbit | rn = 0.529 × n² Å (= n² × a₀, where a₀ = 0.529 Å) |
| Velocity in nth orbit | vn = 2.18 × 10⁶/n m/s |
| Energy of nth level | En = −13.6/n² eV |
Ground state (n=1): E = −13.6 eV; First excited (n=2): E = −3.4 eV
3. Hydrogen Spectral Series
1/λ = R(1/n₁² − 1/n₂²) where R = 1.097 × 10⁷ m⁻¹ (Rydberg constant)
| Series | Transition to n₁ | Region |
|---|---|---|
| Lyman | n₁ = 1 | Ultraviolet |
| Balmer | n₁ = 2 | Visible |
| Paschen | n₁ = 3 | Infrared |
| Brackett | n₁ = 4 | Infrared |
| Pfund | n₁ = 5 | Infrared |
Solved Examples
Example 1
Find the energy of the electron in the 3rd orbit of hydrogen.
Answer: E₃ = −13.6/3² = −13.6/9 = −1.51 eV
Example 2
Find the wavelength of the first line of the Balmer series (n₂ = 3 to n₁ = 2).
Answer: 1/λ = R(1/4 − 1/9) = R(9−4)/36 = 5R/36 = (5 × 1.097 × 10⁷)/36 = 1.524 × 10⁶
λ = 1/(1.524 × 10⁶) = 6.56 × 10⁻⁷ m = 656 nm (red light)
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Quick Revision Points
- Bohr: quantised orbits, L = nh/(2π), En = −13.6/n² eV
- rn ∝ n²; vn ∝ 1/n; En ∝ −1/n²
- Ground state: n=1, E = −13.6 eV; ionisation energy = 13.6 eV
- Lyman (UV), Balmer (visible), Paschen/Brackett/Pfund (IR)
- 1/λ = R(1/n₁² − 1/n₂²)
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Alpha-Scattering & Rutherford’s Nucleus
Geiger and Marsden fired alpha particles at gold foil and a few bounced straight back, revealing a tiny dense nucleus.
Nucleus ~10⁻15 m; whole atom ~10⁻10 m (≈105× bigger).
- Most α pass undeflected → atom is mostly empty space.
- ≈1 in 8000 rebound >90° → mass + positive charge sit in a tiny nucleus.
- Flaw: an orbiting electron should radiate, spiral in → classical atom is unstable.
Distance of Closest Approach
In a head-on hit the alpha’s kinetic energy fully converts to electrostatic PE, fixing how close it gets.
Upper estimate of nuclear size; r0 ∝ 1/K (faster α → closer).
- Set K = (1/4πε0)(2e)(Ze)/r0 at the turning point.
- α charge is +2e — keep the factor 2.
- Convert MeV → joules (1 MeV = 1.6×10⁻13 J) before plugging in.
Impact Parameter
How far off-centre the alpha is aimed (b) sets how sharply it scatters.
b = 0 (head-on) → θ = 180° (perfect rebound).
- Small b (near head-on) → large scattering angle θ.
- Large b → small deflection, nearly straight through.
- The bullseye is a tiny target, so rebounds are rare.
Bohr’s Three Postulates
Bohr bolted quantum rules onto Rutherford’s atom to make it stable and explain line spectra.
Works for H and one-electron ions (He⁺, Li2⁺); fails for multi-electron atoms.
- Stationary orbits: electrons revolve without radiating, though accelerating.
- Angular momentum is quantised in units of h/2π (n = 1,2,3…).
- Light emitted/absorbed only during a jump: down → emit, up → absorb.
Radius of the nth Orbit
Coulomb pull as centripetal force plus quantised L gives discrete orbit sizes.
n=1 hydrogen value 0.529 Å is the Bohr radius; r ∝ n2/Z.
- Radius grows as n2 — outer orbits are far larger.
- Higher nuclear charge Z pulls the electron in tighter.
- He⁺ (Z=2) ground radius = 0.529/2 = 0.265 Å.
Speed in the nth Orbit
The orbiting electron slows down as it moves to higher orbits.
H ground state v ≈ c/137; 1/137 is the fine-structure constant.
- Speed v ∝ Z/n — Z in the numerator, n in the denominator.
- Higher n → slower electron (and longer de Broglie wavelength).
- Independent of where you measure: fixed per orbit.
Energy of the nth Orbit
Total energy is negative because the electron is bound to the nucleus.
H ground state −13.6 eV → ionisation energy = 13.6 eV.
- E ∝ Z2/n2; negative sign means bound.
- Energy bookkeeping: K = −E, U = 2E = −2K, E = U/2.
- First excited state of H (n=2): E = −13.6/4 = −3.4 eV.
Rydberg Formula & Spectral Series
Electron jumps to a lower level emit photons of fixed wavelength, grouped into named series.
Lyman (n_f=1) UV; Balmer (n_f=2) visible; Paschen/Brackett/Pfund IR.
- First line of a series = longest λ, least energy (smallest jump).
- Series limit (n_i = ∞) = shortest λ of that series.
- Cascading from level n emits n(n−1)/2 distinct lines.
de Broglie’s Standing Wave
The electron is a wave that must close on itself, which derives Bohr’s quantisation rule.
Full-wavelength condition (not half-wave); nth orbit holds exactly n wavelengths.
- Only a whole number of wavelengths fits → orbits are discrete.
- Turns Bohr’s assumed L = nh/2π into a derived result.
- As n rises, electron slows so λ grows (λ ∝ n).
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Frequently Asked Questions
Firing alpha particles at a thin gold foil showed most passed nearly undeflected while about 1 in 8000 bounced back beyond 90 degrees. This proved that almost all the mass and the entire positive charge of an atom sit in a tiny central nucleus about 10 to the power minus 15 metres across, with the rest of the atom being mostly empty space.
The orbit radius is r_n equal to 0.529 times n squared over Z angstrom, the speed is v_n equal to 2.18 times 10 to the 6 times Z over n metres per second, and the energy is E_n equal to minus 13.6 times Z squared over n squared electron volts. Angular momentum is quantised as mvr equal to n times h over 2 pi.
The Rydberg formula is 1 over lambda equal to R times Z squared times the quantity 1 over n_f squared minus 1 over n_i squared, where R is 1.097 times 10 to the 7 per metre. Only the Balmer series, with the electron falling to n_f equal to 2, lies in the visible region; the Lyman series is ultraviolet and the Paschen, Brackett and Pfund series are infrared.
Yes, Atoms is part of the NEET Physics syllabus and is consistently asked, usually one question per year. Most questions are direct numericals on Bohr-orbit radius, speed, energy, or hydrogen-spectrum wavelengths, so memorising the standard formulas reliably scores marks.
The distance of closest approach is how near a head-on alpha particle gets to the nucleus before its kinetic energy fully converts to electrostatic potential energy, and it gives an upper estimate of nuclear size. The impact parameter is the perpendicular off-centre distance of the alpha’s initial path from the nucleus; a small impact parameter gives a large scattering angle while a large one gives almost no deflection.