Atomic spectra
Every element has a barcode
Pass the light from an ordinary bulb through a prism. You get a smooth band of all colours, from red to violet, with no gaps. This is a continuous spectrum. A hot, dense solid like the filament gives it.
Now take a glass tube filled with a gas at low pressure, and pass an electric discharge through it. The gas glows. Hydrogen glows pinkish red, neon glows orange-red, sodium vapour glows yellow.
Send this light through the prism. Instead of a rainbow, we see a few bright, sharp lines on a dark background. This is an emission line spectrum.
Each line is light of one particular wavelength. Hydrogen gives a red line at 656.3 nanometre, a blue-green one at 486.1, and violet ones at 434.0 and 410.2.
Sodium gives a bright yellow pair near 589 nanometre. That is the colour of sodium street lamps. Mercury gives strong blue, green and yellow lines.
Every element has its own set of lines, and no two elements share the same set. The line spectrum is a fingerprint, or a barcode, for the element.
So spectroscopy can tell what a sample is made of, just by looking at its light. A few lines are enough to identify the element.
Here is the puzzle. Why does an atom emit only certain wavelengths, and not a continuous range? Classical physics had no answer.
Dark lines and the hydrogen series
Now do the opposite. Send white light through a cool sample of the gas, and then through the prism. The rainbow appears, but with some dark lines cut across it. This is an absorption spectrum.
Put it under the emission spectrum of the same gas. The dark lines sit exactly where the bright lines were. A gas absorbs the very wavelengths it emits.
The spectrum of sunlight has many such dark lines. They tell us which elements are in the Sun’s outer layers. Helium was first found this way, in the Sun, before it was found on Earth.
Look closely at hydrogen’s visible lines. Going from red towards violet, they get closer and closer together, crowding towards a limit at 364.6 nanometre. Such a regular set of lines is called a spectral series.
In 1885 Balmer found a simple formula that fits them: 1 over λ equals R times, 1 over 2² minus 1 over n², with n equal to 3, 4, 5 and so on. R is the Rydberg constant, 1.097 × 10⁷ per metre.
Let us check n equal to 3. 1 over 4 minus 1 over 9 is 5 over 36. 1.097 × 10⁷ times 5 over 36 is 1.524 × 10⁶ per metre. So λ is 656.3 nanometre: the red line.
As n becomes very large, 1 over n² goes to zero, and 1 over λ becomes R over 4. That gives λ equal to 364.6 nanometre, the series limit.
Hydrogen has other series too, outside the visible. Replace 2² by 1²: the Lyman series, in the ultraviolet. Use 3², 4² and 5²: the Paschen, Brackett and Pfund series, in the infrared.
Why should one simple formula describe all these lines? Rutherford’s atom could not say. The answer came from Niels Bohr, as we will see next.
Try it: read the barcodes
- Pick each gas and compare its lines. Drag across the spectra to read off a wavelength.
- In every gas, the dark absorption lines are exactly where the bright emission lines are.
- For hydrogen, tick “Balmer formula lines”: the formula lands on every line, n = 3, 4, 5 …
- Tick “Mystery gas” and name it from its lines alone.
The second Balmer line
Using Balmer’s formula 1/λ = R(1/2² − 1/n²) with R = 1.097 × 10⁷ m⁻¹, find the wavelength of the hydrogen line for n = 4.
- 656 nm
- 486 nm
- 410 nm
- 365 nm
Answer
B. 1/2² − 1/4² = 1/4 − 1/16 = 3/16. So 1/λ = 1.097 × 10⁷ × 3/16 = 2.057 × 10⁶ m⁻¹, and λ = 486 nm, the blue-green line. 656 nm is the n = 3 line. 365 nm comes from forgetting to square: 1/2 − 1/4 = 1/4 gives λ = 4/R, which is actually the series limit.
White light through sodium
White light is passed through cool sodium vapour and then through a prism. What do you see?
- Bright yellow lines near 589 nm on a dark background
- A continuous rainbow with dark lines near 589 nm
- A continuous rainbow with no change
- A rainbow with dark lines at wavelengths where sodium does not emit
Answer
B. This is an absorption spectrum. The cool vapour removes the wavelengths that sodium itself emits, so the rainbow has dark lines at 589.0 and 589.6 nm, just where sodium’s bright emission lines are. Bright lines on a dark background would be the emission spectrum of glowing sodium.
Summary
- Balmer series (visible):
1/λ = R (1/2² − 1/n²), n = 3, 4, 5, … - Rydberg constant:
R = 1.097 × 10⁷ m⁻¹ - Balmer series limit:
λ = 4/R = 364.6 nm
- A hot solid gives a continuous spectrum; a glowing gas at low pressure gives an emission line spectrum: bright lines of definite wavelengths.
- White light through a cool gas gives an absorption spectrum: dark lines at exactly the wavelengths the gas emits.
- Each element has its own set of lines, a fingerprint used to identify elements, even in the Sun and stars.
- Hydrogen’s lines form series: Lyman (ultraviolet), Balmer (visible), Paschen, Brackett and Pfund (infrared).