Why Diwali Fireworks Glow in So Many Different Colours

No microscope shows the inside of an atom. Its structure was worked out from experiments with electric discharges, fast particles and light — each result forcing a better model.

This part covers subatomic particles, the Thomson and Rutherford models, the nature of light, and Bohr's model of the hydrogen atom. Take $h=6.626\times 10^{-34}$ J s and $c=3\times 10^8$ m/s.

Cathode-ray tubes showed a negatively charged particle common to all matter, the electron; the oil-drop experiment fixed its charge; anode rays revealed positive particles, the lightest being the proton; and a neutral particle of about the proton's mass, the neutron, completes the set.

- Cathode rays travel from the negative electrode in straight lines and bend towards a positive plate, whatever gas or electrode is used — so electrons are in every atom. Their charge-to-mass ratio is $1.76\times 10^{11}$ C/kg
- Oil-drop experiment: charges on tiny oil drops are always whole-number multiples of $1.602\times 10^{-19}$ C, the charge of one electron
- Anode (canal) rays: positive particles whose charge-to-mass ratio depends on the gas; the lightest, from hydrogen, is the proton
- Neutron: no charge, mass slightly more than a proton

Worked example — mass of the electron.

$$m_e=\frac{e}{e/m}=\frac{1.602\times 10^{-19}}{1.76\times 10^{11}}\approx 9.1\times 10^{-31}\text{ kg}$$

With $m_p=1.673\times 10^{-27}$ kg, a proton is about $1.84\times 10^3$ times heavier.

An everyday example. Old box-type television sets used a cathode-ray tube, steering a beam of electrons to light up the screen.

The substance. Cathode rays have the same charge-to-mass ratio for every gas, but anode rays do not — because the positive particles are ions of that gas.

Thomson pictured the atom as a uniform sphere of positive charge with electrons embedded in it, but alpha particles fired at thin gold foil were sometimes deflected sharply, which only a tiny, dense, positive nucleus could cause — so Rutherford placed the positive charge and nearly all the mass in a nucleus with electrons revolving around it.

Scattering evidence:

- Most alpha particles passed straight through — the atom is mostly empty space
- A few were deflected by small angles — they passed near a concentrated positive charge
- A very small fraction bounced almost straight back — they approached a tiny, massive nucleus head-on

Worked example — how small is the nucleus? A nucleus has a radius near $10^{-15}$ m and an atom near $10^{-10}$ m, a ratio of $10^5$. If the nucleus were a $1$ cm marble, the atom would be

$$1\text{ cm}\times 10^5=10^5\text{ cm}=1\text{ km across}$$

Limitations of Rutherford's model: an electron moving in a circle is accelerating, and electromagnetic theory says it should radiate energy and spiral into the nucleus — yet atoms are stable. The model also cannot explain line spectra or how electrons are arranged.

An everyday example. Throwing marbles across an empty cricket ground with a single stump in the middle, almost every marble rolls through; only the rare direct hit bounces back.

The substance. Thomson's model explained why atoms are neutral but could not explain the large deflections.

**Light obeys $c=\nu \lambda$, wavenumber is $\bar{\nu }=\frac{1}{\lambda }$, and each quantum of light carries energy $E=h\nu$; black-body radiation and the photoelectric effect could be explained only by treating light as packets of energy called photons.

Worked example 1 — yellow light.** $\lambda =580$ nm:

$$\nu =\frac{3\times 10^8}{5.8\times 10^{-7}}\approx 5.17\times 10^{14}\text{ Hz}\bar{\nu }=\frac{1}{5.8\times 10^{-7}}\approx 1.72\times 10^6{\text{ m}}^{-1}$$

$$E=h\nu =6.626\times 10^{-34}\times 5.17\times 10^{14}\approx 3.43\times 10^{-19}\text{ J}$$

Black-body radiation. A hot body's colour shifts from red to white as it heats; wave theory could not match this, but Planck's quantum theory — energy absorbed or emitted in quanta of $h\nu$ — could.

Photoelectric effect. Light ejects electrons from a metal only above a threshold frequency, and $\frac{1}{2}m{v}^2=h\nu -W_0$.

An everyday example. Automatic street lights switch on at dusk using a light sensor based on the same idea.

The substance. Brighter light below the threshold frequency ejects no electrons at all — intensity changes how many electrons come out, frequency changes their energy.

**Bohr proposed that electrons move only in fixed orbits of energy $E_n=-2.18\times 10^{-18}\frac{Z^2}{n^2}$ J and radius $r_n=52.9\frac{n^2}{Z}$ pm, and that light is emitted only when an electron drops between orbits, giving lines with $\bar{\nu }=R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$, where $R_H=1.097\times 10^7$ m${}^{-1}$.

Series in hydrogen:** Lyman ($n_1=1$, ultraviolet), Balmer ($n_1=2$, visible), Paschen ($n_1=3$, infrared).

**Worked example 1 — the $n=2$ orbit of hydrogen.**

$$E_2=\frac{-2.18\times 10^{-18}}{4}\approx -5.45\times 10^{-19}\text{ J}{r}_2=52.9\times 4\approx 212\text{ pm}$$

Worked example 2 — the red Balmer line ($n_2=3\to n_1=2$):

$$\bar{\nu }=1.097\times 10^7\left(\frac{1}{4}-\frac{1}{9}\right)\approx 1.52\times 10^6{\text{ m}}^{-1}\lambda \approx 656\text{ nm}$$

Limitations: Bohr's model fails for atoms with more than one electron, cannot explain the splitting of lines in magnetic or electric fields, and assumes an exact orbit — which the uncertainty principle forbids.

An everyday example. Sodium vapour street lamps glow a pure yellow, one of sodium's own spectral lines.

The substance. Orbit energies are negative because the electron is bound; zero energy means it has been removed completely.