Why flames have colours

A flame is not one colour because it is hot. It is coloured because the atoms in it give back energy as light, and each kind of atom can only give it back in fixed portions.

Light in portions

Heat a gas of atoms, or hit it with a spark, and it glows. Pass that glow through a prism and it does not spread into a smooth rainbow. It breaks into a few sharp, separate lines of colour, with darkness between them. Every element has its own set of lines. The set is a fingerprint: it tells you which atoms are glowing.

Why it matters

Those lines were measured long before anyone could say why they were there. Classical physics predicted the opposite: an electron circling a nucleus should radiate continuously, lose energy and spiral inward, so atoms should not even be stable. In 1913 Niels Bohr set that prediction aside and built a model that accounted for the lines. When he calculated the constant that fixes hydrogen's line positions — the Rydberg constant, then one of the most precisely measured numbers in physics — his value matched the measured one closely. That agreement is why quantised energy inside atoms was taken seriously.

Levels, gaps and photons

Bohr's model has two rules. First, an electron in an atom may only have certain energies — a ladder of energy levels, numbered n = 1, 2, 3, … Second, while it sits on a level it emits nothing. It emits or absorbs light only when it jumps between levels, and the light comes as a single photon — a packet whose energy equals the gap it crossed.

The level energies are set by one constant. For hydrogen, level n has energy −k/n², with k = 2.179 × 10⁻¹⁸ J. The minus sign means the electron is bound; zero is the ionization limit, where it has escaped altogether. The rungs crowd together as n grows. For a one-electron ion with nuclear charge Z, such as He⁺, the same formula holds with k multiplied by Z², so the whole ladder stretches.

Atoms are most stable at their lowest energy, so an undisturbed hydrogen electron sits on n = 1: the ground state. A flame or spark lifts it to a higher rung: an excited state. Left alone, it falls back, and the energy difference leaves as a photon. Energy is conserved, so exactly the amount that lifted the electron comes back out. A gap is fixed by two whole numbers, so only certain photon energies are possible — and a photon's energy fixes its wavelength, its colour. Hence the lines.

Not every gap is visible. In the textbook's worked example, hydrogen's jump between n = 4 and n = 6 lies in the infrared. In its second, an electron in He⁺ falling from n = 5 to n = 3 releases 6.198 × 10⁻¹⁹ J at a wavelength of 3.205 × 10⁻⁷ m — ultraviolet, just past violet. The colour a heated element gives off is whichever part of its ladder lands inside the narrow visible band.

Bohr's neat circular orbits did not survive; the model fails for helium, with two electrons, and proper quantum mechanics replaced it. What survived is the part that explains the flame: energies come in levels, and the lines in a spectrum are the gaps between them.

In short

Energy lifts an electron up a ladder of allowed levels; it falls back and releases the difference as one photon. Because the ladder has fixed rungs, only fixed photon energies — fixed colours — can come out. Each element has its own ladder, so each has its own lines, and a flame shows whichever lines land in the visible.