Atoms - Quick Revision
Atomic models
- Thomson (plum pudding): positive charge spread uniformly, electrons embedded in it; whole atom neutral.
- Rutherford (nuclear model): from alpha-particle scattering off gold foil. Most mass and all positive charge sit in a tiny nucleus (~10^-15 to 10^-14 m); electrons orbit at ~10^-10 m. Most of the atom is empty space.
- Distance of closest approach: d = 2Ze^2 / (4 pi e0 K). For a 7.7 MeV alpha on gold (Z=79), d ~ 3.0 x 10^-14 m.
Failures of the classical (Rutherford) atom
- An orbiting (accelerated) electron must radiate, spiral in, and the atom would collapse => predicts instability.
- It would emit a continuous spectrum, not the observed discrete line spectrum.
Bohr model (hydrogenic atoms)
- Postulate 1: electrons revolve in certain stable, non-radiating stationary orbits.
- Postulate 2 (quantisation): angular momentum L = mvr = nh/2pi, n = 1, 2, 3...
- Postulate 3: a jump from Ei to lower Ef emits a photon with h(nu) = Ei - Ef.
- Orbit radius: rn = 0.53 n^2 Angstrom (n=1 is the Bohr radius).
- Energy levels: En = -13.6/n^2 eV. Ground state E1 = -13.6 eV.
- Ionisation energy of ground-state H = 13.6 eV.
Hydrogen spectrum
- Rydberg formula: 1/lambda = R(1/n1^2 - 1/n2^2), R = 1.097 x 10^7 m^-1.
- Lyman series (n1=1, UV): first line 122 nm. Balmer (n1=2, visible): H-alpha 656 nm. Paschen (n1=3, IR).
- Excitation energies: n=1 to 2 needs 10.2 eV; n=1 to 3 needs 12.09 eV.
de Broglie explanation
- An allowed orbit fits a whole number of electron wavelengths: 2 pi rn = n lambda, which gives back mvr = nh/2pi.
Limitations of Bohr's model
- Works only for single-electron (hydrogenic) atoms; fails for helium and beyond.
- Cannot explain relative intensities of spectral lines; inconsistent with the uncertainty principle.