Bohr's model of hydrogen

Bohr kept Rutherford's nucleus but added three postulates for hydrogen:

• The electron can move only in certain allowed orbits, and while in one it does not radiate energy. These are stationary states.
• In an allowed orbit, the angular momentum is a whole-number multiple of h/2π: mvr = nh / (2π), with n = 1, 2, 3, .... n is the principal quantum number.
• The atom gives out (or takes in) energy only when the electron jumps between orbits, as one photon with hν = E_i – E_f.

Combining the quantum condition with the Coulomb force gives everything about each orbit:

Example: n = 2: r = 4 × 0.53 × 10⁻¹⁰ = 2.12 × 10⁻¹⁰ m, and E = –13.6/4 = –3.4 eV.

Why negative? The electron is bound: energy must be given to free it. At n = ∞, E = 0 (free electron). The energy needed to free it from the ground state, 13.6 eV, is the ionisation energy of hydrogen.

Higher orbits are bigger, slower, and have higher (less negative) energy. The levels crowd closer together as n grows.

In any orbit: kinetic energy K = –E and potential energy U = 2E, so K + U = E.

The orbital period grows as n³. For big objects like the Earth round the Sun, n would be about 10⁷⁴, so the steps are far too small to notice, and the motion looks smooth. Bohr's rules matter only on the atomic scale.

Bohr's model works well for hydrogen and hydrogen-like ions (He⁺, Li²⁺), but not for atoms with many electrons. Quantum mechanics later replaced it.