Bohr's Model

In 1913 Bohr broke with classical physics in a few bold assumptions, and out came the hydrogen spectrum, number for number.

Bohr’s postulates#

1. The electron moves round the nucleus in circular orbits.
2. Only orbits of certain radii are allowed. In these stationary orbits the electron does not radiate, despite Maxwell.
3. Each stationary orbit has a definite energy. An electron jumping from energy E₂ to a lower E₁ emits one photon with E₂ − E₁ = hν = hc/λ; it can also absorb energy and jump up.
4. In a stationary orbit the angular momentum is a whole multiple of h/2π: Bohr’s quantization rule.

Orbits and energies#

For hydrogen and hydrogen-like ions (He⁺, Li²⁺, … one electron, nucleus of charge Ze): the Coulomb force gives the centripetal force,

Combining with the quantization rule:

Radii

For hydrogen the smallest orbit is the Bohr radius, a₀ = ε₀h²/πme² ≈ 0.053 nm. In general

Energy levels

E₁ = −13.6 eV, E₂ = −3.4 eV, E₃ = −1.5 eV… The lowest level is the ground state; the others are excited states. Negative energy means the electron is bound; zero is the electron and nucleus far apart.

The spectrum explained#

At room temperature nearly every atom is in its ground state, with nowhere lower to go: no radiation. Supply energy and some electrons jump up; dropping back from level m to level n they emit

Every jump ending on n = 1 gives a line between 121.6 nm (2 → 1) and 91.2 nm (∞ → 1): the Lyman series. Ending on n = 2: 656.3 nm down to 365.0 nm, the Balmer series. Ending on n = 3: 1875 nm to 822 nm, the Paschen series.

Ionization, binding and excitation#

• Give a ground-state hydrogen atom more than 13.6 eV and its energy becomes positive: the electron is free. The minimum is the ionization energy (13.6 eV); the voltage that gives an electron that energy is the ionization potential (13.6 V).
• The binding energy is the energy released in assembling the atom from a far-apart electron and proton: also 13.6 eV for hydrogen.
• The excitation energy of a state is the energy to lift the atom to it from the ground state: 10.2 eV for hydrogen’s first excited state (excitation potential 10.2 V).

Where Bohr’s model falls short#

A huge success in 1913, and still how many people picture atoms. But it couldn’t be extended to atoms with more than one electron (not even helium), and precise measurements show the 656.3 nm line is really at least seven close lines. And its postulates are a patch: electrons are simply excused from obeying electromagnetism in certain orbits.