Discrete emission lines
Why hydrogen does not emit light like a hot object.
When an electric current passes through low-pressure hydrogen gas, the emitted light does not form a continuous spectrum. Instead, hydrogen emits light at a small number of very specific wavelengths, producing a line spectrum.
This behavior is fundamentally different from that of a blackbody emitter. A blackbody produces a continuous distribution of wavelengths that depends only on temperature. Hydrogen, by contrast, emits light only at discrete, sharply defined wavelengths.
No classical model of matter or radiation could explain why hydrogen emits light at only certain wavelengths.
Balmer’s empirical discovery
In 1885, Johann Jakob Balmer discovered a simple mathematical formula that accurately described the visible emission lines of hydrogen. Remarkably, this formula was found without any underlying physical model of the atom.
Balmer showed that the wavelengths of the visible hydrogen lines follow the pattern
\[ \frac{1}{\lambda} = R\left(\frac{1}{2^2}-\frac{1}{n^2}\right), \qquad n = 3,4,5,\dots \]
This set of lines is now called the Balmer series. Each value of \(n\) corresponds to a different emission line.
Energy form of the Balmer formula
Using \(E = hc/\lambda\), Balmer’s result can be rewritten in terms of photon energies:
\[ E = R_H\left(\frac{1}{2^2}-\frac{1}{n^2}\right), \]
where \( R_H = 2.18\times10^{-18}\,\text{J} \) is the Rydberg energy for hydrogen.
This equation correctly predicts the energies of the photons emitted by hydrogen atoms in the visible region of the spectrum.
Why this result was so striking
Balmer’s formula works extraordinarily well, yet it was purely empirical. It was not derived from any model of atomic structure, forces, or electron motion.
Balmer himself emphasized that the simplicity of the formula suggested a deeper physical reason for its success, even though such a reason was unknown at the time.
The existence of discrete emission lines — and the ability to describe them with such a simple mathematical pattern — was a major clue that energy in atoms must be quantized.
Big idea: the hydrogen emission spectrum cannot be explained classically. Balmer’s empirical formula revealed an underlying order that would later demand a quantum-mechanical explanation.
Your turn
Key points (one glance)
- Hydrogen emits light at discrete wavelengths, producing a line spectrum.
- This behavior is fundamentally different from blackbody radiation.
- Balmer found an empirical formula describing the visible hydrogen lines.
- The Balmer series corresponds to transitions ending at \(n=2\).
- The energy form is \( E = R_H\left(\frac{1}{2^2}-\frac{1}{n^2}\right) \).
- \( R_H = 2.18\times10^{-18}\,\text{J} \) for hydrogen.
- The simplicity of the formula suggested an underlying physical explanation.
- Explaining why Balmer’s formula works requires quantum theory.