type: worked_problem | subtopic: Numerical Problem Solving

Problem 1: Bohr Model — n=3 Orbit of Hydrogen

Given: Z=1 (hydrogen), n=3. Find r_{3}, v_{3}, $E_{3}$, K$E_{3}$, P$E_{3}$.

Solution: $r_3 = a_0 \frac{n^2}{Z} = 0.529 \times \frac{9}{1} = 4.761\ \text{Å} = 4.761 \times 10^{-10}\ \text{m}$ $v_3 = \frac{2.18 \times 10^6 \times Z}{n} = \frac{2.18 \times 10^6 \times 1}{3} = 7.27 \times 10^5\ \text{m/s}$ $E_3 = -\frac{13.6 Z^2}{n^2} = -\frac{13.6 \times 1}{9} = -1.511\ \text{eV}$ $KE_3 = -E_3 = +1.511\ \text{eV}$ $PE_3 = 2E_3 = -3.022\ \text{eV}$ Verification: $KE_{3}$ + $PE_{3}$ = 1.511 + (−3.022) = −1.511 eV = $E_{3}$ ✓

Compared to ground state (n=1): r_{3}/r_{1} = 9, v_{3}/v_{1} = 1/3, $E_{3}$/$E_{1}$ = 1/9.