Atoms & Nuclei — Bohr Model & Spectra (Medium)

Quick Recap — Bohr Model & Spectra

• Energy levels (hydrogen-like): $E_n=-13.6\dfrac{Z^2}{n^2}$ eV; radius $r_n=0.529\dfrac{n^2}{Z}$ Å; speed $v_n\propto\dfrac{Z}{n}$; angular momentum $L=\dfrac{nh}{2\pi}$.
• Energetics: $KE=-E_n$, $PE=2E_n$, total $=E_n$ (so $KE=-TE$). Ionisation energy of H (from ground state) $=13.6$ eV.
• Photons: transition energy $\Delta E=E_{high}-E_{low}$; wavelength $\lambda\text{(nm)}=\dfrac{1240}{\Delta E\text{(eV)}}$.
• Series: Lyman ($\to n=1$, UV), Balmer ($\to n=2$, visible), Paschen ($\to n=3$, IR). A de-excitation from level $n$ can give $\dfrac{n(n-1)}{2}$ spectral lines.
• de Broglie in orbit: $2\pi r_n=n\lambda$.

Worked mini-example. The $n=2\to1$ transition in hydrogen releases $\Delta E=-3.4-(-13.6)=10.2$ eV, giving $\lambda=\dfrac{1240}{10.2}\approx122$ nm (a Lyman line, in the UV).