Bohr's Model of the Atom and Its Limitations

Niels Bohr (1913) built the first atomic model to correctly reproduce the observed hydrogen line spectrum, by grafting a single quantum assumption onto an otherwise classical picture of an electron orbiting the nucleus under electrostatic attraction. Bohr's model rests on the following postulates:

1. Stationary orbits. The electron in a hydrogen atom can move only in certain fixed circular orbits of definite radius and energy, called stationary states or orbits. As long as the electron stays in one of these orbits, it does not radiate energy, even though classical electromagnetic theory says an accelerating (orbiting) charge should continuously lose energy as radiation and spiral into the nucleus. This postulate is what keeps the atom stable.

2. Quantisation of angular momentum. Only those orbits are allowed for which the angular momentum of the electron is an integral multiple of $h/2\pi$:

where $m$ is the electron's mass, $v$ its speed, $r$ the orbit radius, $h$ Planck's constant, and $n$ (the principal quantum number) labels the orbit. This single condition, together with the balance of electrostatic attraction and centripetal requirement, fixes the radius and energy of every allowed orbit: $r_n = 52.9\, n^2/Z\ \text{pm}$ and $E_n = -2.18 \times 10^{-18}\, Z^2/n^2\ \text{J}$ for a one-electron species of nuclear charge $Z$.

3. Emission and absorption of energy. An electron can jump from one allowed orbit to another. When it falls from a higher orbit (energy $E_2$) to a lower one (energy $E_1$), the atom emits a photon of energy exactly equal to the difference, $\Delta E = E_2 - E_1 = h\nu$; to jump the other way, from a lower to a higher orbit, the atom must absorb a photon of exactly that same energy. Because the orbit energies are discrete (not continuous), only photons of certain sharp frequencies can be emitted or absorbed -- which is exactly what produces a line spectrum rather than a continuous one, and Bohr's formula for $\Delta E$ reproduces the observed hydrogen spectral series (Lyman, Balmer, Paschen, …) essentially exactly. …