The Bohr Model: How Hydrogen's Light Was Decoded

In 1913 Niels Bohr published a model of the hydrogen atom that broke the rules of classical physics on purpose Bohr 1913^[1]. It was wrong in important ways, and it was replaced within a dozen years. It also predicted the wavelengths of hydrogen's light from nothing but the electron's charge and mass and Planck's constant Bohr model, Wikipedia^[2]. Our Niels Bohr sticker is a nod to the man; the numbers are below.

What problem was Bohr trying to solve?

Rutherford's scattering experiments had shown that an atom has a tiny, heavy, positive nucleus with electrons around it Bohr model, Wikipedia^[2]. Classical electrodynamics says that this cannot last. An orbiting electron is accelerating, and an accelerating charge radiates power, as Joseph Larmor showed in 1897 Bohr model, Wikipedia^[2]. The electron should lose energy and spiral into the nucleus. Real atoms are stable, and when hydrogen glows it emits only certain sharp wavelengths. Johann Balmer had fitted the visible lines with an empirical formula in 1885 Hydrogen spectral series, Wikipedia^[3], and Rydberg had generalized it, but nobody could say where the formula came from Bohr model, Wikipedia^[2].

What are Bohr's two rules?

Rule 1: only some orbits are allowed

Bohr postulated that the electron can move only in certain stationary orbits, in which it does not radiate Bohr model, Wikipedia^[2]. For circular orbits, the condition is that the electron's angular momentum is a whole-number multiple of the reduced Planck constant Bohr model, Wikipedia^[2]:

m_e·v·r = n·ħ (n = 1, 2, 3, …)

where m_e is the electron mass in kg, v its speed in m/s, r the orbit radius in m, ħ = h/2π the reduced Planck constant in J·s, and n the principal quantum number.

Rule 2: light is emitted only in jumps

Radiation is emitted or absorbed only when the electron jumps from one stationary orbit to another, and the photon carries away the energy difference, hν = E_upper − E_lower Bohr model, Wikipedia^[2].

What energies and sizes does the model predict?

Combine Rule 1 with Coulomb attraction and Newton's second law and the allowed energies follow Bohr model, Wikipedia^[2]:

E_n = −R_y / n², R_y ≈ 13.606 eV

where R_y is the Rydberg unit of energy. Its CODATA value is 13.605 693 122 990 eV Rydberg constant, Wikipedia^[4], so the ground state of hydrogen sits 13.6 eV below a free electron. The radii grow as n² Bohr model, Wikipedia^[2]:

r_n = n²·a₀, a₀ ≈ 5.292 × 10⁻¹¹ m

where a₀ is the Bohr radius, 5.291 772 105 44(82) × 10⁻¹¹ m Bohr radius, Wikipedia^[5]. For a nucleus of charge Ze, energies scale by Z² and radii by 1/Z Bohr model, Wikipedia^[2]. In the ground state the electron's speed works out to αc, where α ≈ 1/137 is the fine-structure constant, about 2.19 × 10⁶ m/s, under 1% of the speed of light (our arithmetic).

How does it explain the spectral lines?

Put the energy levels into Rule 2 and divide by hc. The result is the Rydberg formula Bohr model, Wikipedia^[2] Hydrogen spectral series, Wikipedia^[3]:

1/λ = R_H·(1/n₁² − 1/n₂²)

where λ is the wavelength in m, n₁ < n₂ are the lower and upper levels, and R_H is the Rydberg constant for hydrogen in m⁻¹. Bohr's derivation expresses this constant in terms of the electron's charge and mass and Planck's constant Bohr model, Wikipedia^[2]. The modern value for an infinitely heavy nucleus is R_∞ = 10 973 731.568 157(12) m⁻¹ Rydberg constant, Wikipedia^[4].

Dividing by 1 + m_e/M corrects for the nucleus's finite mass M Rydberg constant, Wikipedia^[4]. For hydrogen that gives R_H ≈ 1.096 78 × 10⁷ m⁻¹. Then n₁ = 2 and n₂ = 3 gives 656.47 nm in vacuum (our arithmetic), the red H-alpha line, tabulated as 656.3 nm in air Hydrogen spectral series, Wikipedia^[3]. The jumps that end on n = 2 make the Balmer series, the steps on our Balmer Ladder poster. The jumps that end on n = 1 make the ultraviolet Lyman series, starting at 121.57 nm Hydrogen spectral series, Wikipedia^[3].

Why did the reduced mass matter so much?

The nuclear-mass correction was more than a detail. Singly ionized helium has one electron around a nucleus of charge 2e, so its lines should be hydrogen's scaled by Z² = 4. The measured ratio was not exactly 4. It matched 4 times the ratio of the reduced-mass factors, which helped convince Rutherford that Bohr's model mattered Bohr model, Wikipedia^[2].

What does the Bohr model get wrong?

• Ground-state angular momentum. The model gives L = ħ for the ground state. Experiment shows it is zero Bohr model, Wikipedia^[2].
• Fine structure. Michelson and Morley resolved hydrogen lines into close pairs in 1887 Fine structure, Wikipedia^[6]. Bohr suggested in 1914 that relativistic precession of the orbits was the cause, and Arnold Sommerfeld gave a successful relativistic formula in 1916 Fine structure, Wikipedia^[6]. Electron spin and the Zeeman effect lie outside the model Bohr model, Wikipedia^[2].
• More than one electron. The model is nearly exact only for two charges orbiting slowly, such as hydrogen, singly ionized helium, doubly ionized lithium or positronium Bohr model, Wikipedia^[2].
• Consistency. The Bohr–Sommerfeld extension was fundamentally inconsistent and led to many paradoxes. The model was replaced by the quantum-mechanical treatment of hydrogen that Wolfgang Pauli first gave in 1925 using Heisenberg's matrix mechanics Bohr model, Wikipedia^[2].

Even so, the energies E_n = −R_y/n² survive unchanged in the Schrödinger solution of hydrogen, apart from the small corrections above. The precision measurement on our Hydrogen 1S–2S poster carries the same numbers into modern physics. For the wider story, see who started quantum mechanics and why two electrons cannot share a state.

Quick FAQ

Is the Bohr model still taught?

Yes, as a first picture of quantized energy levels. Quantum mechanics replaces orbits with orbitals.

Why don't electrons in Bohr orbits radiate?

In the model, because Bohr said so: it was a postulate. Quantum mechanics later explained why stationary states do not radiate.

What evidence supported it early on?

Beyond hydrogen's spectrum, the Franck–Hertz experiment provided early support for discrete energy levels Bohr model, Wikipedia^[2].

References

1. N. Bohr (1913), “I. On the constitution of atoms and molecules,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 26, 1–25 (DOI 10.1080/14786441308634955; metadata verified via Crossref). https://doi.org/10.1080/14786441308634955
2. Wikipedia, “Bohr model”. https://en.wikipedia.org/wiki/Bohr_model
3. Wikipedia, “Hydrogen spectral series”. https://en.wikipedia.org/wiki/Hydrogen_spectral_series
4. Wikipedia, “Rydberg constant”. https://en.wikipedia.org/wiki/Rydberg_constant
5. Wikipedia, “Bohr radius”. https://en.wikipedia.org/wiki/Bohr_radius
6. Wikipedia, “Fine structure”. https://en.wikipedia.org/wiki/Fine_structure

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.