Atoms

The chapter that opens the atom — compact, derivation-rich and reliably worth 4-6 Board marks plus 1-2 questions in JEE/NEET. The story begins with Thomson's plum-pudding model (1898) and the puzzle of line spectra: rarefied gases emit only discrete wavelengths, a 'fingerprint' tied to atomic structure. Geiger and Marsden's alpha-scattering experiment (1911) — 5.5 MeV alphas on gold foil, most passing straight through but 1 in 8000 bouncing back beyond 90 degrees — forces Rutherford's nuclear model: all positive charge and most mass packed into a ~$10^{-15}$ m nucleus, $10^4$-$10^5$ times smaller than the atom. The model's classical machinery ($\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2} = \frac{mv^2}{r}$, E = -K = U/2 = $-\frac{e^2}{8\pi\varepsilon_0 r}$) works beautifully — until electromagnetism demands the accelerating electron spiral into the nucleus in nanoseconds, radiating a continuous spectrum. Bohr's three postulates (1913) rescue the atom: stationary orbits, angular momentum quantised as $L = nh/2\pi$, photons emitted only in jumps with $h\nu = E_i - E_f$. Out fall the quantised radii $r_n = n^2 a_0$ ($a_0$ = 0.529 angstrom), energies $E_n = -13.6/n^2$ eV, the hydrogen ionisation energy, and — via the Rydberg formula — every spectral series from Lyman to Pfund. De Broglie's standing-wave argument ($2\pi r_n = n\lambda$) then explains where the mysterious quantisation comes from, before the model's honest limitations (hydrogenic atoms only, no intensities) hand the baton to quantum mechanics. Every derivation, ratio trick and exam trap is covered.