Chapter 12 - Atoms

Rutherford shot alpha particles at thin gold foil. Most passed straight, some deflected small angle, very few bounced back 180 degrees. This meant atom is mostly empty space with tiny dense positive nucleus at centre containing mass.

Atom size ~10^-10 m, nucleus ~10^-14 m - like football stadium with marble at centre. Electrons orbit around like planets but Rutherford model failed because accelerating charge should radiate and fall into nucleus.

Distance of closest approach: alpha with kinetic energy K cannot go closer than r0 = (1/4pi epsilon0) * 2Ze^2 / K where Z gold atomic number.

Bohr fixed Rutherford by adding quantum rules: Electrons revolve in fixed circular orbits called stationary orbits with fixed energy. No radiation in these orbits so no collapse.

Condition 1: Centripetal force = Coulomb attraction: mv^2/r = (1/4pi epsilon0) Ze^2 / r^2.

Condition 2: Angular momentum quantized: mvr = n h/2pi where n=1,2,3... So only certain radii allowed: rn = n^2 a0 / Z where a0 = 0.53 angstrom = Bohr radius. Velocity vn = 2.2 x10^6 / n m/s for hydrogen. Energy En = -13.6 Z^2 / n^2 eV.

Hydrogen energy En = -13.6 / n^2 eV. n=1 -13.6 eV ground state most stable. n=2 -3.4 eV first excited, n=3 -1.51 eV etc. At n infinity 0 eV free electron.

Ionization energy needed to remove electron from ground: 13.6 eV. Excitation energy to go n1 to n2: E2-E1 positive.

When electron jumps down it emits photon h nu = Ei - Ef. When jumps up absorbs photon same energy.

Rydberg formula: wavenumber 1/lambda = R (1/n1^2 - 1/n2^2) where R=1.097x10^7 m^-1. n2 greater than n1.

Lyman series: jumps to n1=1 n2=2,3... UV region ~91 to 121 nm. Balmer: to n1=2 n2=3,4... visible 364 to 656 nm, H alpha 656 nm red famous. Paschen: to n1=3 n2=4... infrared 820 to 1875 nm. Brackett to n1=4, Pfund to n1=5 both IR.

Series limit when n2 infinity gives shortest lambda: lambda_limit = 1/(R/n1^2). Lyman limit 91.2 nm.

de Broglie explained why angular momentum quantized. Electron orbit circumference must fit integer wavelengths for standing wave: 2 pi rn = n lambda = n h/p = n h / mv. Rearranging gives mvr = n h/2pi Bohr condition. So Bohr quantization arises from wave nature.

For n=1 one wavelength fits around circle, n=2 two waves etc. Like guitar string fixed both ends only integer half waves.

Excitation: electron absorbs photon or collision energy and jumps to higher n. Atom excited short lived ~10^-8 s then falls back emitting photon. Excitation energy E_exc = E2-E1 e.g., n=1 to 2 needs 10.2 eV.

Ionization: electron gets enough to reach n infinity free. Energy needed ionization energy 13.6 eV from ground. After ionization atom becomes positive ion.

Franck Hertz experiment proved quantized energy by shooting electrons through mercury vapour. Current dropped at 4.9 eV multiples because electrons lost energy to excite mercury atoms only at fixed levels.

Bohr model works only for hydrogen like one electron atoms. Fails for multi-electron, cannot explain fine structure spectral lines splitting into close lines, Zeeman effect splitting in magnetic field, Stark effect in electric field.

Refinements: Sommerfeld added elliptical orbits introduced azimuthal quantum number l. Later quantum mechanics gave 4 quantum numbers n principal size, l shape, ml orientation, ms spin. Bohr orbits replaced by probability clouds orbitals.

Atomic spectra used in flame test - sodium yellow, potassium violet etc. Sodium lamps yellow street lights use sodium excitation. Neon signs different gases different colours. Lasers work on stimulated emission population inversion between energy levels.

Astronomy identifies elements in stars by absorption lines. Hydrogen 21 cm line used in radio astronomy.