Atoms — Notes

• Thomson model: positive charge spread uniformly through the atom, electrons embedded; predicts only tiny deflections of alpha particles.
• Geiger–Marsden experiment: a collimated beam of alpha particles (about 5.5 MeV) on a very thin gold foil; a ZnS screen and microscope count the particles scattered through each angle θ; apparatus in vacuum.
• Observations: most pass undeflected or nearly so; a few are deflected through large angles; about 1 in 8000 through more than 90°; some almost straight back. The number scattered falls steeply with θ (∝ 1/sin⁴(θ/2)).

• All the positive charge and nearly all the mass are in a tiny nucleus (about $10^{-15}$–$10^{-14}$ m); the atom (about $10^{-10}$ m) is mostly empty space; electrons revolve round the nucleus.
• Impact parameter $b$: perpendicular distance of the initial path from the nucleus. Small $b$ → large deflection; $b = 0$ → $\theta = 180°$; large $b$ → little deflection.
• Distance of closest approach (head-on): $K = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2Ze^2}{d}$, so $d = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2Ze^2}{K}$; an upper limit to the nuclear size.
• Drawbacks: an orbiting (accelerating) electron should radiate, lose energy and spiral into the nucleus; the model predicts a continuous spectrum, not lines.

Postulates: (1) the electron moves in certain orbits without radiating; (2) in them $mvr = \dfrac{nh}{2\pi}$; (3) a jump from $E_i$ to $E_f$ emits a photon with $h\nu = E_i - E_f$.

From $\dfrac{mv^2}{r} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{e^2}{r^2}$ and $mvr = \dfrac{nh}{2\pi}$:

• $v_n = \dfrac{e^2}{2\varepsilon_0 nh} \approx \dfrac{2.18 \times 10^{6}}{n}$ m s⁻¹
• $r_n = \dfrac{n^2h^2\varepsilon_0}{\pi me^2} \approx 0.53n^2$ Å (Bohr radius $r_1 \approx 5.3 \times 10^{-11}$ m)
• $E_n = -\dfrac{me^4}{8\varepsilon_0^2h^2n^2} = -\dfrac{13.6}{n^2}$ eV; $K = -E_n$, $U = 2E_n$.
• Ground state −13.6 eV; ionisation energy 13.6 eV; first excited state −3.4 eV.
• Bohr’s second postulate follows if a whole number of de Broglie wavelengths fits round the orbit: $2\pi r = n\lambda$.
• Works for hydrogen-like atoms only; does not explain many-electron spectra or line intensities.

• A jump from $n_i$ to $n_f$ emits $h\nu = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)$ eV; $\lambda = hc/\Delta E$ ($hc \approx 1240$ eV nm). Discrete levels → line spectrum; absorption lines at the same wavelengths.
• Series: Lyman ($n_f = 1$, ultraviolet, limit ≈ 91 nm); Balmer ($n_f = 2$, visible: 3 → 2 red ≈ 656 nm); Paschen ($n_f = 3$, infrared); Brackett, Pfund (far infrared).
• Lines in a series crowd towards the series limit ($n_i \to \infty$).
• From level $n$, the number of possible lines is $n(n - 1)/2$.