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Every JEE and NEET aspirant meets the Bohr model, usually as a set of formulas to memorise. It is more interesting than that. It was a bold patch on classical physics that worked brilliantly for one atom and failed for the rest, and that failure helped create quantum mechanics. This post covers what Bohr proposed, a worked example, where the model breaks, and what you can take from the way he worked.
Who was Niels Bohr?
Bohr was born in Copenhagen on 7 October 1885 and died there on 18 November 1962, according to his Nobel Prize facts page. After his doctorate he spent years abroad, including in Manchester and Cambridge, then returned to lead Copenhagen University's Institute for Theoretical Physics, now the Niels Bohr Institute, which became a centre for the development of quantum physics. He received the 1922 Nobel Prize in Physics "for his services in the investigation of the structure of atoms and of the radiation emanating from them."
What problem was Bohr trying to solve?
In 1911, Ernest Rutherford concluded from alpha-particle scattering that an atom's positive charge sits in a tiny nucleus. Our post on Rutherford's gold foil experiment tells that story. But classical physics said an electron circling a nucleus should radiate energy continuously and spiral inwards. Atoms would collapse, and they would give off a smear of colours. Real hydrogen gives sharp lines at fixed wavelengths.
What did Bohr propose in 1913?
Bohr's paper On the constitution of atoms and molecules, published in the Philosophical Magazine in 1913, combined Rutherford's nucleus with quantum ideas. In the form you study today:
The electron moves only in certain allowed orbits, where its angular momentum is a whole-number multiple of h/2π: L = nh/2π, with n = 1, 2, 3 and so on.
In an allowed orbit it does not radiate.
When it jumps from a higher orbit to a lower one, it emits one photon with energy equal to the difference: hν = E_upper − E_lower.
For hydrogen this gives energy levels Eₙ = −13.6 eV / n² and orbit radii rₙ = n² × 0.0529 nm. The NIST values are 13.605 693 eV for the energy constant and 5.291 77 × 10⁻¹¹ m for the Bohr radius.
As the Nobel facts page puts it, the model explained why atoms emit light only at fixed wavelengths.
Worked example: the red line of hydrogen
Question. Find the wavelength of light emitted when hydrogen's electron falls from n = 3 to n = 2.
E₃ = −13.6 / 9 = −1.511 eV. E₂ = −13.6 / 4 = −3.400 eV.
Photon energy: ΔE = E₃ − E₂ = −1.511 − (−3.400) = 1.889 eV. Or directly: 13.6 × (1/4 − 1/9) = 13.6 × 5/36 = 1.889 eV.
Wavelength: λ = hc / ΔE. Using hc ≈ 1240 eV nm, λ = 1240 / 1.889 ≈ 656 nm.
That is red light. The measured line, listed in NIST's table of hydrogen lines, is at 6562.85 Å, or about 656.3 nm. A simple model built on two assumptions lands within a fraction of a nanometre.
Common mistakes:
Sign errors. Energies are negative because the electron is bound. Subtract so the photon energy comes out positive.
Using n instead of n². Energy goes as 1/n² and radius as n².
Wrong units. If you use hc = 1240 eV nm, energy must be in eV and wavelength comes out in nm.
Forgetting Z. For hydrogen-like ions such as He⁺, Eₙ = −13.6 Z² / n² eV, so He⁺'s ground state is −54.4 eV.
Where does the Bohr model fail?
Your NCERT Class 12 chapter on atoms lists the key limits. The model works for hydrogen-like atoms but cannot be extended even to helium, with two electrons, because it ignores the forces between electrons. And while it predicts the frequencies of hydrogen's lines, it cannot explain their relative intensities.
There is a deeper problem. Bohr's orbits are sharp paths, like planets. Quantum mechanics later showed that an electron does not have a definite path in this sense.
What replaced it?
Two new theories arrived within a year of each other. In 1925, Werner Heisenberg published his quantum-theoretical reinterpretation of kinematics and mechanics, the start of matrix mechanics. In 1926, Erwin Schrödinger published Quantisation as an eigenvalue problem, introducing the wave equation. Both give the same hydrogen energy levels as Bohr, and much more besides. The orbits became orbitals, described by the quantum numbers n, l, m and s. Our guide to quantum numbers and atomic structure picks up from there.
What can a student learn from how Bohr worked?
Bohr did not wait for a complete theory. He took two things he trusted, Rutherford's nucleus and the sharp lines of hydrogen, and made the smallest bold assumption that fitted both. Then he tested it with numbers anyone could check.
Use the same habit in your own study: when a concept does not fit, write down exactly which fact it fails to explain, then test your idea on a number. If picturing orbits and energy levels is hard, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read.
Practice routine for this week: calculate the n = 4 → 2 and n = 5 → 2 wavelengths yourself, check them against about 486 nm and 434 nm, then do two He⁺ problems from a past paper.
