Bohr's model of the hydrogen atom: the three postulates
Bohr kept the nucleus but added quantum rules: electrons move in stationary orbits without radiating, with angular momentum nh/2π, and emit a photon only when jumping between orbits.
Bohr's three postulates (1913)
- Stationary orbits: an electron can revolve only in certain fixed orbits. In these orbits it does not radiate energy, even though it is accelerating. Each orbit has a fixed energy, so these are called stationary states.
- Quantum condition: the allowed orbits are those in which the angular momentum of the electron is a whole-number multiple of h/2π: mvr = nh/2π, n = 1, 2, 3, ...
- Frequency condition: an electron can jump from an orbit of higher energy to one of lower energy. It then emits a photon whose energy equals the energy difference: hν = E(initial) − E(final). It absorbs a photon of the same energy to jump up.
How the postulates fix Rutherford's problems
Niels Bohr, a Danish physicist, mixed classical mechanics with Planck's quantum idea. Postulate 1 simply declares the allowed orbits stable, so the atom does not collapse. Postulate 2 picks out only some orbits, so the electron can have only certain energies. Postulate 3 then means only certain photon energies can come out. Fixed photon energies mean fixed wavelengths, so the atom gives a line spectrum, as seen in experiments.
de Broglie's explanation of the second postulate
Bohr gave no reason why angular momentum should come only in steps of h/2π. In 1923, Louis de Broglie explained it with the wave nature of the electron.
- 1An electron moving with momentum mv has a de Broglie wavelength λ = h/(mv).
- 2In a stable orbit, the electron wave must join up with itself all round the circle and form a standing wave, like the waves on a plucked string.
- 3So a whole number of wavelengths must fit into the circumference of the nth orbit: 2πrₙ = nλ, n = 1, 2, 3, ...To draw: a circle with a wave running round it that joins smoothly; for n = 4, four whole waves fit.
- 4Put λ = h/(mvₙ): 2πrₙ = nh/(mvₙ).
- 5Rearrange: mvₙrₙ = nh/2π. So, Bohr's quantum condition follows from the wave nature of the electron: only orbits that hold whole standing waves can exist.
Worked example: angular momentum in an orbit
Find the angular momentum of an electron in the third orbit (n = 3) of hydrogen. By how much does it change when the electron jumps to n = 2? (h = 6.63 × 10⁻³⁴ J s, π = 3.14)
- Formula: L = nh/2π.
- h/2π = 6.63 × 10⁻³⁴ ÷ 6.28 = 1.056 × 10⁻³⁴ J s.
- L₃ = 3 × 1.056 × 10⁻³⁴ = 3.17 × 10⁻³⁴ J s.
- Change from n = 3 to n = 2: ΔL = (3 − 2) × h/2π = 1.06 × 10⁻³⁴ J s, a decrease.
L₃ ≈ 3.17 × 10⁻³⁴ J s; it decreases by 1.06 × 10⁻³⁴ J s
Worked example: frequency of the emitted photon
An electron in a hydrogen atom jumps from an orbit of energy −1.51 eV to an orbit of energy −3.40 eV. Find the energy and frequency of the photon emitted. (h = 6.63 × 10⁻³⁴ J s, 1 eV = 1.6 × 10⁻¹⁹ J)
- Formula: hν = E(initial) − E(final).
- Photon energy = −1.51 eV − (−3.40 eV) = 1.89 eV.
- In joules: 1.89 × 1.6 × 10⁻¹⁹ = 3.02 × 10⁻¹⁹ J.
- ν = 3.02 × 10⁻¹⁹ J ÷ 6.63 × 10⁻³⁴ J s = 4.56 × 10¹⁴ Hz.
Photon energy 1.89 eV; frequency ≈ 4.56 × 10¹⁴ Hz (red light)
Limitations of Bohr's model
- It works only for hydrogen-like (hydrogenic) atoms, which have a single electron: H, He⁺, Li²⁺. It fails even for helium, which has two electrons, because it cannot handle the forces between electrons.
- It gives the frequencies of the spectral lines correctly but cannot explain their relative intensities (why some lines are brighter than others).
The colours of Diwali fireworks come from metal salts: strontium gives red, barium green, sodium yellow. Each colour is a photon given out when electrons jump between fixed energy levels, just as Bohr's third postulate says.
2 marks: “Using de Broglie's hypothesis, explain Bohr's second postulate of the quantisation of angular momentum.” The orbiting electron behaves as a standing wave, so a whole number of de Broglie wavelengths fit in its orbit: 2πr = nλ. With λ = h/mv, this gives mvr = nh/2π.
3 marks: “State the three postulates of Bohr's model of the hydrogen atom.” (1) Electron moves in certain stationary orbits without radiating. (2) Allowed orbits have mvr = nh/2π. (3) A jump between orbits emits or absorbs a photon with hν = E(initial) − E(final).
2 marks: “Give two limitations of Bohr's model of the atom.” It applies only to one-electron (hydrogen-like) atoms. It cannot explain the relative intensities of spectral lines.
Orbit, Quantise, Jump: stable Orbits, Quantised L = nh/2π, a Jump gives hν = E(initial) − E(final).