10-Sentence Summary: Atoms & Nuclei

(1) Rutherford's alpha-scattering experiment proved the atom is mostly empty space with a tiny dense nucleus (~10^{-15} m), quantified by the distance of closest approach d = 2k$Ze^{2}$/KE_α.

(2) Bohr's model resolved the instability of the nuclear atom by postulating fixed stationary orbits, quantized angular momentum L = nℏ, and photon emission only during transitions.

(3) In the Bohr model, orbital radius r_n = 0.529$n^{2}$/Z Å increases as $n^{2}$, and total energy E_n = −13.6$Z^{2}$/$n^{2}$ eV becomes less negative (less tightly bound) with higher n.

(4) The kinetic energy KE = −E_n is always positive; potential energy PE = 2E_n is always negative and double the magnitude; their sum equals total energy.

(5) The Rydberg formula 1/λ = R(1/n_{1}^{2} − 1/n_{2}^{2}) gives spectral wavelengths; Lyman series lies in UV (n_{1}=1), Balmer in visible (n_{1}=2), and Paschen in infrared (n_{1}=3).

(6) The number of distinct spectral lines emitted from level n is n(n−1)/2.

(7) Nuclear radius follows R = 1.2 × A^(1/3) fm, making nuclear density constant (~$2.3 \times 10^{17}$ kg/$m^{3}$) for all nuclei regardless of mass number.

(8) Mass defect $\Delta m$ = [Zm_p+(A−Z)m_n]−M converts to binding energy BE = $\Delta m$×931.5 MeV; Fe-56 has the maximum BE/A (~8.75 MeV/nucleon) and is the most stable nucleus.

(9) Both fusion (light nuclei → heavier) and fission (heavy nucleus → medium fragments) release energy because products have higher BE/A than reactants.

(10) Radioactive decay law N = $N_{0}$e^(−λt) gives half-life t_{1}/{2} = 0.693/λ and mean life τ = 1/λ = 1.443t{1}/_{2}; alpha decay reduces A by 4 and Z by 2, beta-minus increases Z by 1 (A unchanged), gamma leaves A and Z unchanged.