The Bohr Model & Hydrogen Spectra
Electrons in an atom can only occupy specific fixed energy levels, like rungs on a ladder — they can never sit in between. When an electron jumps down from a higher rung to a lower one, it emits a single photon carrying exactly that energy difference, which is why hydrogen glows in sharp, distinct colors instead of a smooth rainbow.
The formula
1/λ = R · (1/n₁² − 1/n₂²)
- λ — wavelength (m): the wavelength of light given off by the electron's jump
- R — Rydberg constant (1/m): a fixed constant that sets the scale of hydrogen's energy levels
- n₁ — lower level (—): the energy level the electron lands on, closer to the nucleus
- n₂ — upper level (—): the energy level the electron starts from, farther from the nucleus
Worked example
An electron in a hydrogen atom drops from energy level n = 3 to energy level n = 2 — the first line of the Balmer series. Find 1/λ in terms of the Rydberg constant R.
- n₁ = 2
- n₂ = 3
- 1/λ = R · (1/n₁² − 1/n₂²)
- 1/λ = R · (1/4 − 1/9) = R · (9/36 − 4/36)
1/λ = 5R/36
Test yourself
Which photon carries more energy: an electron dropping from n = 2 to n = 1 (Lyman), or from n = 3 to n = 2 (Balmer)?
- Correct answer: Lyman (n = 2 → 1)
- Balmer (n = 3 → 2)
- They carry the same energy
Right! Dropping all the way to n = 1 is a bigger energy jump than dropping to n = 2, so the Lyman photon has more energy — that's why it lands in the UV instead of visible light.
For the transition n = 2 to n = 1 (the first Lyman line), what is 1/λ in terms of R?
- R/4
- Correct answer: 3R/4
- 3R/2
Yes! 1/n₁² − 1/n₂² = 1/1 − 1/4 = 3/4, so 1/λ = 3R/4 — this is the Lyman-alpha line.
Where you see this
A hydrogen discharge tube glows a specific lavender-pink rather than a smooth rainbow, and astronomers read a star's composition from the dark lines its atoms punch in its spectrum. Both come from electrons dropping between fixed rungs: each drop emits one photon of exactly that rung-to-rung energy.
Common mistakes
First, which jump wins: a bigger fall releases a higher-energy photon — dropping to n = 1 (the Lyman series) out-emits dropping to n = 2 (Balmer), because the rungs bunch up as you climb. Then the arithmetic of the rungs: the photon's 1/λ subtracts inverse SQUARES, so for n = 2 to n = 1 it is 1/1² − 1/2² = 1 − 1/4 = 3/4, giving 1/λ = 3R/4 — not the plain difference of the n's.
How it connects
Energy quantization goes atomic: photons from the photoelectric effect now explain why atoms emit sharp lines at all. But Bohr's rungs were postulated, not explained — matter waves, next, supply the why (allowed orbits are standing waves), which is why this model is taught as the brilliant stepping stone it was.