Hydrogen's emission spectrum is a set of sharp lines that fall into families, or series, each named for its discoverer. With the Bohr energies, every line has a simple label: the level the electron starts in and the level it lands in.
🔑 Key idea
A drop from level $n_i$ to a lower level $n_f$ emits a photon with $$\frac{1}{\lambda} = R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right),\qquad R_H = 1.097 \times 10^7\ \text{m}^{-1}.$$ The series are set by $n_f$: Lyman ($n_f = 1$, ultraviolet), Balmer ($n_f = 2$, visible and near UV), Paschen ($n_f = 3$, infrared). Letting $n_i \to \infty$ gives each series' shortest wavelength, its limit.
💡 Why it works
The photon carries away the energy difference: $\frac{hc}{\lambda} = E_{n_i} - E_{n_f} = 13.6\ \text{eV}\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$. Divide by $hc$: $R = \frac{13.6\ \text{eV}}{1240\ \text{eV·nm}} = 0.01097\ \text{nm}^{-1} = 1.097 \times 10^7\ \text{m}^{-1}$. Johann Balmer found the pattern of the visible lines by trial in 1885; Bohr's model explained it. Because $n_i > n_f$, the bracket is positive, and the lines of a series crowd together toward the limit as $n_i$ grows.
Hydrogen's levels with three Balmer jumps (to $n = 2$) and the first Lyman jump (to $n = 1$). The Lyman jump is far larger, so its photon is ultraviolet.
Hydrogen's first three series, computed with $R_H = 1.097 \times 10^7$ m⁻¹ (vacuum wavelengths).
Series
$n_f$
Region
Longest line
Limit
Lyman
1
ultraviolet
121.5 nm ($2 \to 1$)
91.2 nm
Balmer
2
visible, near UV
656.3 nm ($3 \to 2$)
364.6 nm
Paschen
3
infrared
1875 nm ($4 \to 3$)
820.4 nm
Worked examples
✏️ Worked example: A Balmer line
Find the wavelength of the photon emitted when hydrogen drops from $n = 4$ to $n = 2$.
$\frac{1}{9} - \frac{1}{n_i^2} = 0.0711$, so $\frac{1}{n_i^2} = 0.0400$ and $n_i = 5$.
Answer: $n_i = 5$ (the jump $5 \to 3$)
Astronomers use these lines every day. The red Balmer line (656 nm) gives glowing hydrogen clouds around young stars their pink-red color, and the Lyman-alpha line (121.6 nm, ultraviolet), stretched by the expansion of the universe into visible or infrared light, is one of the main ways the most distant galaxies are found.
⚠️ Common mistake
Swapping $n_f$ and $n_i$. Writing $\frac{1}{n_i^2} - \frac{1}{n_f^2}$ with $n_i > n_f$ gives a negative wavelength; using the wrong level as $n_f$ puts the line in the wrong series. The lower level, where the electron lands, sets the series and goes first in the bracket. Check: Balmer lines must come out between about 365 and 656 nm.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 2 XP.