Master Formula Sheet

# Result Formula
1 Coulomb force (alpha-nucleus) F=14πε02Ze2r2F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2Ze^2}{r^2}
2 Closest approach d=2Ze24πε0Kd = \dfrac{2Ze^2}{4\pi\varepsilon_0 K}; scales as qZK\dfrac{qZ}{K}
3 Force balance e24πε0r2=mv2r\dfrac{e^2}{4\pi\varepsilon_0 r^2} = \dfrac{mv^2}{r}
4 Energy triple K=−EK = -E, U=2EU = 2E, E=−e28πε0rE = -\dfrac{e^2}{8\pi\varepsilon_0 r}
5 Quantisation L=mvrn=nh2πL = mvr_n = \dfrac{nh}{2\pi}
6 Radii rn=n2Za0r_n = \dfrac{n^2}{Z}a_0; a0=5.29×10−11a_0 = 5.29 \times 10^{-11} m
7 Speeds vn=c137Znv_n = \dfrac{c}{137}\dfrac{Z}{n}; v1v_1(H) = 2.18×1062.18 \times 10^6 m/s
8 Energies En=−13.6Z2n2E_n = -13.6\dfrac{Z^2}{n^2} eV
9 Periods Tn∝n3Z2T_n \propto \dfrac{n^3}{Z^2}; T1T_1(H) = 1.53×10−161.53 \times 10^{-16} s
10 Photon rule hν=Ei−Efh\nu = E_i - E_f; λ=1240E(eV)\lambda = \dfrac{1240}{E(\text{eV})} nm
11 Rydberg 1λ=RZ2(1nf2−1ni2)\dfrac{1}{\lambda} = RZ^2\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right); R = 1.097×1071.097 \times 10^7 m−1^{-1}; 1R=91.2\dfrac{1}{R} = 91.2 nm
12 Line count max lines from level n: n(n−1)2\dfrac{n(n-1)}{2}
13 de Broglie fit 2πrn=nλ2\pi r_n = n\lambda; λn=2πna0∝n\lambda_n = 2\pi na_0 \propto n
14 Magnetic moment μn=neh4πm\mu_n = n\dfrac{eh}{4\pi m} (n Bohr magnetons)

Numbers to lock in: -13.6, -3.4, -1.51, -0.85 eV; excitations 10.2, 12.09, 12.75 eV; wavelengths 121.6, 91.2, 656, 486, 365, 1875 nm; nucleus 10−1510^{-15}-10−1410^{-14} m vs atom 10−1010^{-10} m; 30 fm closest approach (7.7 MeV alpha on gold).

Three Models, One Story

Thomson (1898) Rutherford (1911) Bohr (1913)
Positive charge spread through atom tiny nucleus tiny nucleus
Electrons embedded, static orbiting, classical orbiting, quantised
Explains scattering? no (no backscatter) yes yes
Stable? no (electrostatic) no (radiative spiral) yes (stationary states)
Line spectra? no no (continuous) yes (hν=Ei−Efh\nu = E_i - E_f)
Fails at alpha scattering stability & spectra multi-electron atoms, intensities

Geiger-Marsden in three statistics: most alphas undeviated (atom mostly empty); ~0.14% beyond 1 degree (strong fields are rare); ~1 in 8000 beyond 90 degrees (charge + mass concentrated in a tiny nucleus). Impact parameter b: small b ↔ large angle; b = 0 ↔ rebound (θ ≈ π).

Spectral series table:

Series nfn_f Region First line Limit
Lyman 1 UV 121.6 nm 91.2 nm
Balmer 2 Visible 656 nm (Hα\alpha) 365 nm
Paschen 3 IR 1875 nm 820 nm
Brackett 4 IR 4051 nm 1459 nm
Pfund 5 IR 7460 nm 2279 nm

Series limit energy = binding energy of the floor level (13.6, 3.4, 1.51, 0.85 eV…). Cold-gas absorption shows Lyman only.

Scaling Rules & Exam Traps

The exponent family (hydrogen-like, nuclear charge Z):

  • Radius: ∝n2Z\propto \dfrac{n^2}{Z} — squares up, Z down.
  • Velocity: ∝Zn\propto \dfrac{Z}{n} — linear both ways.
  • Energy magnitude: ∝Z2n2\propto \dfrac{Z^2}{n^2} — squares both ways.
  • Period: ∝n3Z2\propto \dfrac{n^3}{Z^2} — the Kepler cube.
  • de Broglie wavelength in orbit: ∝nZ\propto \dfrac{n}{Z}.
  • Orbiting mass m enters as: r∝1mr \propto \dfrac{1}{m}, E∝mE \propto m (muonic-atom questions).

Trap list (each costs marks yearly):

  • 'First excited state' = n = 2; 'second excited state' = n = 3.
  • Ionisation energy depends on the starting level: 13.6 eV only from n = 1.
  • V0V_0-style confusion: excitation energy (eV) vs excitation potential (V).
  • Photon absorption needs an EXACT level match; electron-beam excitation needs only 'at least'.
  • Wavelength ratios: brackets only, R cancels. λ∝1/Z2\lambda \propto 1/Z^2 across ions.
  • Line counts: sample lines n(n−1)2\dfrac{n(n-1)}{2}; single-atom cascade photons ≤ n - 1; per-series lines = (n - floor).
  • L = nh/2π: convert any given L to this form before identifying n.
  • Closest approach: d∝qZKd \propto \dfrac{qZ}{K} — projectile charge matters (proton vs alpha).

One-Glance Revision Flow

The chapter in eight steps:

  1. Thomson (1898): uniform positive pudding + embedded electrons; same atomic size as later models; electrostatically unstable; cannot explain scattering.
  2. Geiger-Marsden (1911): 5.5 MeV alphas on gold; mostly through, 1 in 8000 backscattered → tiny massive positive nucleus (10−1510^{-15}-10−1410^{-14} m), atom mostly empty.
  3. Closest approach: d=2Ze24πε0Kd = \dfrac{2Ze^2}{4\pi\varepsilon_0 K} = 30 fm for gold — an upper bound on nuclear size.
  4. Classical orbits: Coulomb = centripetal; K = -E, U = 2E; r = 0.53 angstrom, v = 2.2×1062.2 \times 10^6 m/s from E = -13.6 eV — but accelerating electrons must radiate: spiral collapse + continuous spectrum. Model dead.
  5. Bohr's postulates: stationary orbits; L=nh/2πL = nh/2\pi; hν=Ei−Efh\nu = E_i - E_f.
  6. Consequences: rn=n2a0r_n = n^2a_0; En=−13.6/n2E_n = -13.6/n^2 eV; ionisation 13.6 eV; excitations 10.2 / 12.09 / 12.75 eV; levels crowd toward 0.
  7. Spectra: Rydberg formula; Lyman UV / Balmer visible / Paschen-Brackett-Pfund IR; limits = floor-level binding energies; n(n−1)2\dfrac{n(n-1)}{2} lines.
  8. de Broglie: 2πrn=nλ2\pi r_n = n\lambda derives the quantisation — standing electron waves; limitations: hydrogenic only, no intensities → quantum mechanics.

Morning-of-exam checklist: -13.6/n² … r ∝ n²/Z, v ∝ Z/n, T ∝ n³/Z² … K = -E, U = 2E … 10.2 eV first excitation, 13.6 eV ionisation … 121.6 / 656 / 91.2 / 365 nm anchors … brackets not R for ratios … n(n-1)/2 lines … cold absorption = Lyman only … d ∝ qZ/K … 2πr = nλ. Go score.