Class 12 · Chapter 12

Atoms

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Atoms Short Notes

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Condensed revision points for Atoms — for quick recall before exams, not a substitute for the full notes.

Thomson's model: positive charge spread through the whole sphere, electrons embedded in it ('plum pudding'). Couldn't explain spectra or large-angle alpha scattering.

Alpha particles fired at thin gold foil; most pass straight through, a few deflect strongly, ~1 in 8000 bounce back.
N(θ) ∝ cosec⁴(θ/2) — sharply peaked at small angles.
Conclusion: atom is mostly empty space, with nearly all mass and all positive charge concentrated in a tiny nucleus.
Nuclear diameter < ~10⁻¹⁵ m; atomic diameter ~10⁻¹⁰ m.

r₀ = (1/4πε₀)(2Ze²)/E_K — head-on closest approach, all KE converted to PE.
b = (1/4πε₀)(Ze²/E_K) cot(θ/2) — small b ⟹ large deflection; large b ⟹ small deflection.

Classically, an accelerating (orbiting) electron should radiate energy continuously and spiral into the nucleus — predicts atoms can't be stable.
Gives no reason for discrete spectral lines instead of a continuous spectrum.

Coulomb force = centripetal force: (1/4πε₀)(Ze²/r²) = mv²/r.
Angular momentum quantized: mvr = nh/2π, n = 1,2,3,...
Electron in an allowed orbit does not radiate — stationary states, constant energy.
Radiation emitted/absorbed only on a jump between orbits: E₂ − E₁ = hν.

r_n = ε₀n²h²/(πmZe²); r_n ∝ n²/Z; r₁(H) = 0.529 Å.
v_n = Ze²/(2ε₀nh); v_n ∝ Z/n; v₁(H) ≈ 2.2×10⁶ m/s.
E_n = −13.6 Z²/n² eV; E_n ∝ Z²/n².
KE = −E_n (always positive); PE = 2E_n (always negative) — PE is twice the magnitude of total energy.
Time period T_n ∝ n³/Z; orbital frequency f_n ∝ Z²/n³; angular momentum L_n = nh/2π ∝ n.

n=1: −13.6 eV (ground state, K-shell). n=2: −3.4 eV (L). n=3: −1.51 eV (M). n=4: −0.85 eV. n=∞: 0 eV.
Binding energy at level n = |E_n|.
First excitation energy (H) = 10.2 eV; ionisation energy (H, ground state) = 13.6 eV.

Excitation: electron jumps to a higher orbit. Ionisation: electron removed completely (n→∞).
Excited states last ~10⁻⁸ s before the electron falls back, emitting a photon.
Emission spectrum: bright lines on dark background (electron drops down). Absorption spectrum: dark lines on bright background (electron jumps up) — same wavelengths as emission, opposite contrast.

Same Bohr formulas apply to any single-electron ion of nuclear charge Ze: He⁺ (Z=2), Li²⁺ (Z=3), etc.
Ionisation energy scales as Z²: He⁺ = 54.4 eV, Li²⁺ = 122.4 eV.
Z of an unknown hydrogen-like ion = √(measured IE / 13.6 eV).

Standing-wave condition: 2πr_n = nλ, with λ = h/(mv).
Combining the two reproduces Bohr's quantization rule directly — orbits are where the electron's wave closes on itself constructively.

Works only for single-electron systems — fails for multi-electron atoms (no electron-electron interaction).
Can't explain fine structure, Zeeman effect, or Stark effect.
Still assumes a precise circular orbit — replaced by probability clouds in true quantum mechanics.
Semi-classical: quantizes energy/angular momentum but keeps a classical circular-motion picture.

Don't forget the negative sign on E_n — 'energy increases' with n means E_n becomes less negative (closer to zero), not larger in magnitude.
KE = −E_n and PE = 2E_n — a very common source of sign and factor-of-2 errors; PE is NOT just −E_n.
Number of spectral lines from n down to 1 is n(n−1)/2, not n² or n−1 — don't forget this combinatorial formula.
Z² appears in energy/ionisation formulas, but Z¹ (not Z²) appears in the velocity formula, and 1/Z in the radius formula — don't apply the same power of Z to every quantity.
Balmer series is the only one with visible lines; all of Paschen, Brackett, and Pfund are infrared — a frequently tested distinction.

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