Understand that isolated atoms, such as atomic hydrogen, possess discrete electron energy levels, and that transitions between these levels give rise to characteristic line spectra.
The Bohr model assumes that an electron moves in a circular orbit around the nucleus with quantised angular momentum:
$$L = m_e v r = n\hbar,\qquad n = 1,2,3,\dots$$Balancing the centripetal force with the Coulomb attraction gives:
$$\frac{m_e v^{2}}{r} = \frac{k e^{2}}{r^{2}}$$Eliminating $v$ using the angular momentum condition leads to the allowed radii:
$$r_{n} = \frac{n^{2}\hbar^{2}}{k m_{e} e^{2}} = n^{2}a_{0},$$ where $a_{0} = \dfrac{\hbar^{2}}{k m_{e} e^{2}} \approx 5.29\times10^{-11}\,\text{m}$ is the Bohr radius.The total energy of the electron in the $n$‑th orbit is the sum of kinetic and potential energy:
$$E_{n} = -\frac{k e^{2}}{2r_{n}} = -\frac{k^{2} m_{e} e^{4}}{2\hbar^{2}}\frac{1}{n^{2}} = -\frac{13.6\ \text{eV}}{n^{2}}.$$When an electron transitions from a higher level $n_{i}$ to a lower level $n_{f}$, a photon is emitted with energy equal to the difference between the two levels:
$$\Delta E = E_{n_{f}} - E_{n_{i}} = hu = \frac{hc}{\lambda}.$$Conversely, absorption of a photon of the same energy promotes the electron to the higher level.
The wavelength of the emitted or absorbed photon can be expressed using the Rydberg constant $R_{\infty}$:
$$\frac{1}{\lambda}=R_{\infty}\!\left(\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}\right),\qquad R_{\infty}=1.097\times10^{7}\ \text{m}^{-1}.$$| Series | Final level $n_{f}$ | Region of spectrum |
|---|---|---|
| Lyman | 1 | Ultraviolet |
| Balmer | 2 | Visible |
| Paschen | 3 | Infrared |
| Brackett | 4 | Infrared |
| Pfund | 5 | Infrared |
The existence of discrete lines in emission and absorption spectra provides direct evidence that electrons can only occupy certain energy states. This quantisation is a fundamental postulate of quantum mechanics and underlies the stability of atoms.