The Bohr Model, Line by Line
By Lucian — builder & engineer, LK Forge
Heat hydrogen and it does not glow in a smear of colour — it emits a handful of razor-sharp lines, always in the same places. In 1913 Niels Bohr explained them with one bold idea: the electron can only sit on a ladder of fixed energies, and light comes out when it jumps between rungs. We took the Bohr lab’s formula, checked its colours against real hydrogen, and pulled a fundamental constant out of the slope.
· 6 min read · exact from Eₙ = −13.6/n²
The four colours of hydrogen
This is what you see through a spectroscope pointed at a hydrogen tube: a black band crossed by four bright lines, nothing in between. Each is one Balmer jump, placed by the model exactly where the real line sits. The same four lines, formed in the atmospheres of stars, are how we know the universe is mostly hydrogen.
The Balmer series on a 380–740 nm strip. Bohr places them at 656.5 / 486.3 / 434.2 / 410.3 nm; the real hydrogen lines are 656.3 / 486.1 / 434.0 / 410.2 — a match to two or three parts in ten thousand.
One slope, one constant
Every hydrogen line, in every series, obeys 1/λ = R·(1/n_lo² − 1/n_hi²). Plot the inverse wavelength against that bracket for jumps drawn from the Lyman, Balmer and Paschen series and they fall on a single straight line through the origin. Its slope is the Rydberg constant — the same number whether the line is ultraviolet, visible or infrared.
Seven transitions (dots) from three series, all on one line whose slope is R = 1.097×10⁷ m⁻¹ — because R is nothing but 13.6 eV divided by hc.
Three series, three bands of light
Which rung the electron lands on decides the colour of the whole family. Only the Balmer series (landing on n = 2) falls in visible light — an accident of hydrogen’s energy spacing that made it the first spectrum ever decoded.
| Series | Lands on | First line | Band | Series limit |
|---|---|---|---|---|
| Lyman | n = 1 | 121.6 nm | ultraviolet | 91.2 nm |
| Balmer | n = 2 | 656.5 nm | visible | 364.7 nm |
| Paschen | n = 3 | 1875.6 nm | infrared | 820.6 nm |
The series limit (n_hi → ∞) is where the lines crowd together and stop — the ionisation edge. Balmer’s limit at 364.7 nm sits just past the violet, at the edge of sight.
Reproduce it yourself
The whole spectrum is two lines of arithmetic; the third pulls out the Rydberg constant.
const E = n => -13.6 / (n * n); // eV
const dE = (hi, lo) => 13.6 * (1/(lo*lo) - 1/(hi*hi)); // eV
const lambda = (hi, lo) => 1240 / dE(hi, lo); // nm
lambda(3, 2); // 656.5 nm Hα (red) real 656.3
lambda(4, 2); // 486.3 nm Hβ (cyan) real 486.1
// Rydberg constant = slope of (1/lambda) vs (1/lo^2 - 1/hi^2):
const R = (1 / (lambda(3,2) * 1e-9)) / (1/4 - 1/9); // 1.097e7 /mThe energy-level and wavelength formulas are lifted straight from the Bohr solver; the series table and Rydberg fit are the small script behind this post.
Build a spectrum
Open the lab, pick a series, and fire transitions one by one to watch each line appear on the spectrum exactly where the ladder puts it.