• $summary_{type}$: 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^2$$a_0$/Z, where $a_0$ = 0.529 Angstrom is the Bohr radius. Velocity: $v_n$ = Z$v_0$/n, where $v_0$ = 2.18 x 10^6 m/s = c/137 (the fine structure constant appears naturally). Energy: $E_n$ = -13.6*$Z^2$/$n^2$ eV. The negative sign indicates a bound state. The virial theorem gives: KE = -$E_{total}$ = 13.6$Z^2$/$n^2$ eV (positive), PE = 2*$E_{total}$ = -27.2$Z^2$/$n^2$ 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.