• summary_type: concept
• word_count: 180

From Bohr's postulates, combining the centripetal force equation (mv^2/r = kZe^2/r^2) with angular momentum quantization (mvr = n-hbar), we derive all orbital quantities. Radius: r_n = n^2a_0/Z, where a_0 = 0.529 Angstrom is the Bohr radius. Velocity: v_n = Zv_0/n, where v_0 = 2.18 x 10^6 m/s = c/137 (the fine structure constant appears naturally). Energy: E_n = -13.6Z^2/n^2 eV. The negative sign indicates a bound state. The virial theorem gives: KE = -E_total = 13.6Z^2/n^2 eV (positive), PE = 2E_total = -27.2Z^2/n^2 eV. So KE = |E| and PE = -2|E|, or equivalently PE = -2*KE. Time period: T_n proportional to n^3/Z^2. Current: I_n proportional to Z^2/n^3. Angular momentum: L_n = n-hbar (independent of Z — this is often tested). Key scaling: radius grows as n^2, velocity decreases as 1/n, energy becomes less negative as 1/n^2. For Z=1, n=1: r = 0.529 A, v = 2.18 x 10^6 m/s, E = -13.6 eV.