Hydrogen Atomic Spectrum — Bohr/Rydberg Model
Line spectra of hydrogen arise when an electron changes its principal quantum level
\(n=1,2,3,\dots\). The energy of the photon equals the magnitude of the atomic energy change:
\[
E_{\text{photon}} = h\nu = \frac{hc}{\lambda} = \bigl|\Delta E_{\text{atom}}\bigr| .
\]
Energy levels
In the Bohr model (and from solving the Schrödinger equation for H), the level
energies are
\[
E_n = -\,\frac{R_\mathrm{H}hc}{n^2}
\approx -\,\frac{2.179\,872\times10^{-18}\ \mathrm{J}}{n^2}
= -\,\frac{13.6057\ \mathrm{eV}}{n^2}.
\]
Here \(R_\mathrm{H}\) is the Rydberg constant for hydrogen
(\(R_\mathrm{H}\approx 1.096\,776\times10^{7}\ \mathrm{m^{-1}}\)),
\(h\) is Planck’s constant, and \(c\) is the speed of light.
Rydberg relation for wavelengths
For a transition from \(n_i\) to \(n_f\),
\[
\frac{1}{\lambda}
= R_\mathrm{H}\!\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right),\qquad
n_i \ne n_f .
\]
If \(n_i>n_f\) the atom emits a photon (emission line);
if \(n_i
Series and spectral regions
| Series | Final level \(n_f\) | Region (typical) | Famous line(s) |
|---|
| Lyman | 1 | Ultraviolet | \(\lambda_{\alpha}\approx 121.6\ \mathrm{nm}\) (2→1) |
| Balmer | 2 | Visible / near-UV | H-α 656.3 nm (3→2), H-β 486.1 nm, H-γ 434.0 nm |
| Paschen | 3 | Infrared | \(\approx 1875\ \mathrm{nm}\) (4→3) |
| Brackett, Pfund, … | 4, 5, … | Infrared | Longer wavelengths as \(n_f\) increases |
Ionization limits
The ionization limit of a series is obtained by taking \(n_i\to\infty\).
From level \(n_i\) to the continuum \(n_f=\infty\),
\[
\frac{1}{\lambda_\text{limit}} = R_\mathrm{H}\!\left(0-\frac{1}{n_i^2}\right) \;\Rightarrow\;
\lambda_\text{limit} = \frac{n_i^2}{R_\mathrm{H}} .
\]
Example: From the ground state \(n_i=1\), \(\lambda_\text{limit}=1/R_\mathrm{H}\approx 91.2\ \mathrm{nm}\).
What the calculator does
- You enter \(n_i\) and \(n_f\) (or choose ionization, \(n_f=\infty\)).
- It applies the Rydberg relation to find \(\lambda\), then computes
\(\nu=c/\lambda\) and \(E=h\nu\) (per photon; it also reports kJ·mol⁻¹).
- It identifies the spectral series and, if applicable, the approximate visible color.
Notes.
(i) The constants here are for isolated hydrogen (electron–proton); small corrections
from reduced mass, fine structure, or external fields are neglected.
(ii) Wavenumber used in spectroscopy is \(\tilde{\nu}=1/\lambda\) in cm⁻¹
(the tool reports this as well).
