Physics · Ch 13 — Atoms

Summary

Rutherford’s $\alpha$ -particle scattering experiment showed that the atom has a tiny, dense, positively charged nucleus, with electrons orbiting at large distances — most of the atom is empty space.
The nuclear model could not explain the stability of atoms or the discrete line spectra observed; classical physics predicted that accelerating electrons would spiral into the nucleus, emitting a continuous spectrum.
Bohr’s postulates for the hydrogen atom:
- Electrons move only in certain stationary orbits where angular momentum is quantized: $m v r = n \frac{h}{2\pi}$ , $n = 1,2,3,\dots$
- While in these orbits, electrons do not radiate energy.
- Radiation is emitted or absorbed only when an electron jumps between orbits: $h\nu = E_i - E_f$ .
From Bohr’s model, the radius of the $n$ th orbit in hydrogen is $r_n = n^2 a_0$ , where $a_0 = 0.529 \,\text{Å}$ is the Bohr radius.
The energy of the $n$ th level is $E_n = -\frac{13.6}{n^2} \,\text{eV}$ (ground state $n=1$ : $-13.6$ eV). Energy is negative because the electron is bound.
The wavelength of emitted/absorbed light is given by the Rydberg formula: $\frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$ , with $R = 1.097 \times 10^7 \,\text{m}^{-1}$ .
Spectral series of hydrogen:
- Lyman (ultraviolet): $n_f = 1$ , $n_i = 2,3,\dots$
- Balmer (visible): $n_f = 2$ , $n_i = 3,4,\dots$
- Paschen (infrared): $n_f = 3$ , $n_i = 4,5,\dots$
- Brackett and Pfund (far infrared): $n_f = 4$ and $5$ respectively. …

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