The full, readable lecture — how the atom went from a "plum pudding" to a tiny nucleus, why Bohr forced electrons onto fixed quantized orbits, the energy-level ladder Eₙ = −13.6/n², how jumps between rungs emit and absorb photons, the hydrogen line spectrum and its Lyman, Balmer and Paschen series, the Rydberg formula, X-rays, and lasers. As you scroll, the panel on the right plays out each idea with an everyday object you already know — a hailstorm on a sheet, a staircase, a barcode, a torch beam.
1 — From plum pudding to the nuclear atom
By 1900 the atom was a puzzle. J. J. Thomson pictured it as a "plum-pudding" — a smooth ball of positive charge with electrons studded throughout it like raisins. It seemed reasonable, but it was wrong, and one beautiful experiment proved it.
Rutherford's gold-foil experiment (Geiger & Marsden, 1909) fired fast, positively-charged alpha particles at a sheet of gold leaf only a few atoms thick. If the plum-pudding were right, the diffuse charge should barely deflect them.
- Most passed straight through — so the atom is mostly empty space.
- A few were deflected, and about 1 in 8000 bounced almost straight back — "as if you fired a shell at tissue paper and it came back at you."
- Conclusion — the positive charge and nearly all the mass are concentrated in a minute, dense nucleus at the centre; electrons orbit far outside it.
Exam point: the rare large-angle scatter is the key evidence. It requires a tiny, concentrated positive nucleus — impossible for Thomson's spread-out pudding.
2 — The Bohr model: quantized orbits
Rutherford's atom had a fatal flaw: a circling electron should radiate energy and spiral into the nucleus in a fraction of a second. In 1913 Niels Bohr fixed it with a bold postulate — electrons may occupy only certain allowed orbits, and in these stationary states they do not radiate.
Think of a staircase, not a ramp: on a ramp you can stand at any height, but on a staircase you can only stand on a step. An electron is the same — it sits on an allowed orbit or jumps to another, but it can never hover in between.
Bohr's quantum conditionmvr = n · (h / 2π) n = 1, 2, 3, …
angular momentum is quantized in steps of h/2π
- Each orbit is labelled by a whole number n, the principal quantum number.
- An electron in a stationary state has a fixed energy and emits no radiation.
- Radiation appears only when the electron jumps between orbits.
Why "quantized"? Only whole numbers of n are allowed — there is no orbit 1.5. The atom's energies come in discrete steps, which is exactly why its light comes in sharp lines, not a smear.
3 — Energy levels: Eₙ = −13.6/n²
Solving Bohr's model for hydrogen gives the energy of the electron in the n-th orbit. The energies are negative — the electron is bound, trapped in the nucleus's pull — and they climb towards zero as n increases.
Energy of the n-th level (hydrogen)Eₙ = −13.6 / n² eV
E₁ = −13.6 eV · E₂ = −3.40 eV · E₃ = −1.51 eV · E∞ = 0
Picture an energy ladder whose rungs bunch up near the top. The bottom rung (n = 1, the ground state) is deep down at −13.6 eV; each higher rung is closer to its neighbour, crowding together towards 0 eV at the top.
| Level n | Energy Eₙ | Name |
|---|
| 1 | −13.6 eV | ground state |
| 2 | −3.40 eV | first excited |
| 3 | −1.51 eV | second excited |
| ∞ | 0 eV | ionised (free) |
Ionisation energy of hydrogen = the climb from n = 1 to n = ∞ = 0 − (−13.6) = 13.6 eV — the energy to tear the electron free.
4 — Emission & absorption of photons
Light is born when an electron jumps between rungs. When it falls from a higher level to a lower one, the lost energy leaves the atom as a single photon — a flash of light whose colour is fixed by exactly how far it fell.
Bohr's frequency conditionhf = E_high − E_low
E = hf = hc / λ (h = 6.63 × 10⁻³⁴ J·s)
- Emission — electron drops down a rung → atom releases a photon (a flash). Bigger fall → higher frequency → bluer light.
- Absorption — atom swallows a photon of the exact right energy → electron climbs up a rung. The reverse process.
Like stepping down a stair releases a click of sound, an electron stepping down releases a flash of light — and only the exact step energies are allowed, so only certain colours ever appear.
Key idea: the photon's energy must match the gap exactly. A photon that is too big or too small is simply ignored — which is why each element absorbs and emits its own fixed set of colours.
5 — The hydrogen line spectrum
Because only fixed energy gaps exist, hydrogen emits light at only a few sharp wavelengths — a line spectrum, not a continuous rainbow. The lines group into series by the level the electron lands on.
| Series | Electron falls to | Region |
|---|
| Lyman | n = 1 | ultraviolet (UV) |
| Balmer | n = 2 | visible |
| Paschen | n = 3 | infrared (IR) |
The pattern is a barcode fingerprint of colours: every element has its own unique set of lines. Astronomers read the bright and dark lines in starlight to tell exactly which elements a distant star is made of.
Exam point: the Balmer series falls in the visible band, so it is the one you actually see — including the famous red H-alpha line (n = 3 → 2). Lyman is in the UV, Paschen in the IR.
6 — The Rydberg formula
Long before Bohr, Johannes Rydberg found an empirical formula that reproduces every hydrogen line with stunning accuracy. Bohr's model later explained why it works.
Rydberg formula1/λ = R (1/n₁² − 1/n₂²) (n₂ > n₁)
R = 1.097 × 10⁷ m⁻¹ · n₁ = lower level, n₂ = upper level
- n₁ sets the series — n₁ = 1 Lyman, n₁ = 2 Balmer, n₁ = 3 Paschen.
- n₂ is the higher level the electron started in.
- R is the Rydberg constant; 1/λ is the wavenumber.
balmer — the H-alpha line
Find λ for the n = 3 → 2 (Balmer) transition.
1/λ = R (1/2² − 1/3²) = 1.097×10⁷ (0.25 − 0.111)
1/λ = 1.097×10⁷ × 0.1389 = 1.524×10⁶ m⁻¹
λ = 656 nm — red light
7 — X-rays and their uses
X-rays are very short-wavelength, high-energy electromagnetic waves. They are produced in an X-ray tube: a heated filament boils off electrons, a large voltage (tens of kilovolts) accelerates them to enormous speed, and they crash into a metal target (often tungsten).
- How they form — the fast electrons are abruptly stopped by the target; their lost kinetic energy is radiated as X-ray photons.
- Maximum energy — when an electron gives up all its energy at once: hf_max = eV, the accelerating voltage sets the shortest wavelength.
- Properties — they travel in straight lines, are not deflected by fields, and penetrate flesh but not bone or metal.
Shortest-wavelength (cut-off) X-rayeV = hf_max = hc / λ_min ⟹ λ_min = hc / (eV)
Uses: medical imaging of bones and teeth, CT scans, airport security scanners, and crystallography (revealing the structure of DNA and proteins). Over-exposure is harmful — they are ionising radiation.
8 — Lasers & chapter recap
A LASER (Light Amplification by Stimulated Emission of Radiation) makes a beam that is monochromatic (one colour), coherent (all waves in step) and tightly collimated (a thin, parallel beam).
- Stimulated emission — a passing photon nudges an excited atom to emit a second, identical photon: same energy, same direction, same phase. One photon becomes two, two become four — light is amplified.
- Population inversion — to amplify rather than absorb, more atoms must be in the upper state than the lower one. This unnatural condition is set up by pumping energy in.
- Optical cavity — two mirrors bounce the light back and forth, building up an avalanche; one mirror is slightly leaky and lets the beam out.
Uses: CD/DVD/Blu-ray players, barcode scanners, fibre-optic communication, surgery and eye correction, cutting and welding, distance measurement and surveying.
- Gold-foil: most alphas pass, a few bounce back → tiny dense nucleus.
- Bohr: electrons on quantized orbits (a staircase); no radiation in a stationary state.
- Energy levels: Eₙ = −13.6/n² eV; ground state n = 1; ionisation = 13.6 eV.
- Photons: hf = E_high − E_low; emission down, absorption up.
- Hydrogen spectrum: Lyman (UV), Balmer (visible), Paschen (IR) — a fingerprint.
- Rydberg: 1/λ = R(1/n₁² − 1/n₂²).
- X-rays: fast electrons hit a metal target; eV = hc/λ_min.
- Lasers: stimulated emission + population inversion → coherent light.